CD

0A NON-DIMENSIONAL ANALYSIS OF

CARDIOVASCULAR RESPONSE TO COLD STRESS

PART II: DEVELOPMENT OF THE NON-DIMENSIONAL PARAM ETERS

D T IC J" Sharon A. Starowicz, M.S.SELECTE Biomedical Engineer

MAR2 M U.S. Food and Drug Administration

D 7, Silver Spring, Maryland 20910

TD

Daniel J. Schneck, Ph.D.

Professor of Engineering Science and Mechanics

Head, Biomedical Engineering Prograr

Virginia Polytechnic Institute and State University

227 Norris Hall

Blacksburg, Virginia 24061

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11. TITLE (Include Security Classification)

A NON-DIMENSIONAL-ANALYSIS OF CARDIOVASCULAR RESPONSE TO COLD STRESS

PART II: DEVELOPMENT OF THE NON-DIMENSIONAL PARAMETERS [PART I: NMRI 83-51]

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17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

FIELD GROUP SUB-GROUP Hypothermia Dimensionless Parameters

_ _ __ Tempe.ature __. Cardiovascular ControlVascular System Blood

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TABLE OF CONTENTS

Page

Chapter

I Introduction and Statement of the Problem .................... 1

II Historical Background ...................................... 6

III Method of Analysis ......................................... 9

A. The Buckingham Pi Theorem .............................. 9

B. Simplification and Physical Significance ofDimensionless Parameters .............................. 13

IV Tabulated Results of the Analysis .......................... 14

V Discussion of Results ...................................... 18

V!I Conclusion................................................ 25

References......................................................... 29

Appendix.......................................................... 31

Table 1 - Tabulation of Derived Dimensionless Parameters ... 32

A. Blood................................................ 32

B. Heart................................................ 43

C. Vascular System ....................................... 58

D. Miscellaneous........................................ 80

Table 2 - Derived Fundamental Scales Associated with Electro-

thermodynamic Effects ..................................... 86

Table 3 - Other Derived Scales Associated with Electrotherno-

dynamic Effects.......................................... 87

Table 4 - List of Symbols Used in Table 5..................... 88

TABLE OF CONTENTS (continued)

Page

Table 5 - Derived Dimensionless Parameters Written in Termsof Traditional Non-Dimensional Numbers Related to Fluidand Thermodynamic Theory ................................ 91

Table 6 - List of Variables Which Are Already Dimensionlessby Definition .............................................. 99

Table 7 - Other Variables That Can Be Associated with Cardio-vascular Function and Thermoregulation .................. 103

Table 8 - Enzymes ............................................... 107

Table 9 - Hormones .............................................. 09

Table 10 - Nutrients ............................................

Table 11 - Vitamins ............................................. 113

Table 12 - Buffering Ions, Minerals and Electrolytes, TraceElements, Blood Gases, Byproducts of Metabolism ......... 114

Accesion

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CHAPTER I

Introduction and Statement of the Problem

The thermoregulatory function of the cardiovascular system is ex-

tremely complex. It involves numerous physical and chemical parameters

that interact )o maintain a relatively constant temperature within the

body core, an anatomical region containing organs vital to the sustenance

of life.4Jnder conditions of cold, the body utilizes a variety of feed-

back control regulatory mechanisms that attempt both to decrease heat loss

to the environment and increase heat production. The primary effector of

such control is the cardiovascular system, which has the ability to rapid-

ly change blood flow patterns throughout the body in response to this need.

The reaction of the system to a moderately cold stimulus has been

studied fairly extensively and is understood quite well -t 3 Not so

well understood, however, is what happens when one is subjected to ex-

tremes of cold, where body reactions can often be quite adverse. Not

only is performance greatly restricted in the cold environment due to

impaired musculoskeletal function, but a host of pathologic conditions

result from the inability of the system to function under extremely low

temperatures. In fact, the same mechanisms that attempt to protect and

maintain the body in the presence of moderately cool temperatures, may

indeed promote the destruction of the system when it is subjected to

extremes in cold.

Relatively little is known about the physiology of temperature

1|

2

regulation during prolonged periods of cold exposure. The complexities

of the cardiovascular system even under normal operating conditions are

vast. The fluid, blood, possesses certain non-Newtonian, anisotropic,

and viscoelastic properties that directly influence its ability to trans-

port heat. In addition to carrying heat, blood is responsible for trans-

porting various biochemical constituents that are directly involved in

thermoregulatory processes. The properties of blood are also highly

temperature-dependent. For example, as the fluid is cooled it becomes

more viscous and harder to pump, creating a tendency toward decreased

blood flow.

An equally important constituent of cardiovascular thermoregulatory

response is the vascular system, which provides the "pipes" through which

the fluid is pumped. Both heat and mass transport occur across the blood

vessel walls and the ease with which blood can be pumped through them is

directly dependent upon various temperature-related elastic properties of

the vessels themselves. In addition, the blood vessels are able to regu-

late the pathways of blood flow through local vasoconstrictive mechanisms

in response to a cold stress.

The heart serves as the pump for the system and provides the energy

needed to drive the blood. Under continuous nervous control, it can adjust

its pumping rate to change cardiac output as needed.

In addition to these "hardware" components, one must also consider

the intrinsic control mechanisms that govern the overall system response

to any external stimulus. This would include the endocrine system as well

as the central and autonomic nervous systems.

3

Any approach to studying the effects of cold on the cardiovascular

system must necessarily account for all the governing factors mentioned

above. This is to ensure model accuracy, since so many aspects of the

system are interrelated and dependent on one another. The analysis must

seek to isolate as many thermal, mechanical, and chemical properties of

the system as possible, toward the ultimate goal of obtaining a detailed

understanding of how these properties interact to protect body function

S 14 effectively from temperature extremes in assaulting environments. With

Ahis) in mind, the Aorkiwhich follows -escribes a non-dimensional approach

which was taken to describe cardiovascular function

Over six hundred physical and chemical Variables that govern the

thermoregulatory function of the cardiovascular system have been identi-

fied [13], and even this impressive list is far from exhaustive. In this

study, the variables related to properties of blood, the vascular wall,

the heart, and the cardiovascular control system were defined using

dimensions of mass, length, time, temperature, and charge. From these,

a working set of dimensionless parameters was developed using the Buck-

ingham Pi Theorem.(see Chapter III and Table I of Chapter IV). -This

allowed the thermal, mechanical, and chemical properties of the cardio-

vascular system to be coupled through the non-dimensionalization process.

In order to obtain insight into the physics of the problem and to under-

stand what the dimensionless groupings represent, these parameters were

simplified and related to one another through geometric ratios and non-

dimensional numbers common to the fields of fluid mechanics, solid and

fluid heat conduction and convection, viscoelasticity, and electrochemical

4

diffusion (see Tables 4 and 5). On this basis, comparison of the param-

eters in terms of their relative significance to the problem was made.

In addition, various scales were derived that are associated with elec-

trothermodynamic processes and these were used to identify the physical

significance of each parameter (see Tables 2 and 3).

Such a preliminary non-dimensional approach exploits a powerful

technique for recognizing the fundamental aspects of the problem before a

more complicated mathematical formulation is attempted. Because one can

not initially know all the details associated with the response of the

cardiovascular system to cold stress, and because there exist numerous

experimental limitations that inhibit comprehensive study, a non-dimen-

sional analysis provides a first order understanding of physical processes

associated with the problem. This presents a foundation for future

mathematical treatment.

In addition to identifying what experimental parameters might be

useful for modeling studies, this analysis has direct application to the

military, where prediction of physiologic response to adverse combat

situations is often desired. Along these lines, relevant and measurable

design parameters for protective clothing and gear may be proposed to

enhance natural thermoregulation and compensate for adverse environmental

reaction. Going one step further, the development of design parameters

that may be useful in optimizing protective gear, and the suggestion of

training methods to affect conditioning or adaptation to a cold environ-

ment, may all result from studies of this kind. By quantifying the

final parameters and exploring their variability among individuals, a

basis for clinical diagnosis and treatment may be established, as well

as a protocol for predicting one's susceptibility to cardiovascular

stress in a cold environment.

CHAPTER II

Historical Background

The use of dimensional analysis is not recent. The development

of the Buckingham Pi Theorem in 1914 provided a systematic mathematical

approach to determine dimensionless groupings that contained variables

pertinent to a given problem [6]. This technique is exploited in

the following study and is explained in detail in Chapter III.

In 1927, Lambert and Teissier [10] first dealt with the question

of circulatory similitude by proposing that the cardiac cycle and other

biological periods are proportioiial to the length dimension of the body.

Historically, similarity studies of the cardiovascular system have been

based on well-established hydrodynamic principles [10]. However, such

principles have often been used only to define local hemodynamic similar-

ities without regard to the system as a whole. Current trends in this

area combine such traditional methods with modern pulse transmission

theory and cardiac dynamics, in an attempt to establish similarity criter-

ia for the entire system that directly relate the physiological response

of the system to its structural design [10].

A variety of cardiovascular phenomena have been explored using

similarity criteria and dimensional analysis. McComis, Charm, and Kurland

[11] attempted to characterize pulsating blood flow by using dimensional

analysis as a method for obtaining equations which describe various

pressure-velocity relationships. By investigating the parameters that

6

7

mignt influence the pulsating flow of blood through small tubes, and

organizing these through dimensional analysis, they were successful in

characterizing the steady flow of blood through small glass tubes.

Further experimentation allowed the relationship between such parameters

to be expressed in a simple form.

Another area where dimensional analysis has found great applicability

is that of determining optimal design features of the cardiovascular

system. For example, the recent works of Li and Milnor [10] have

suggested that the arterial pulse transmission path length and wavelength

are of such a ratio that the amount of external work performed by the

ventricle at the resting heart rate is at a minimum. Using allometric

equations for the given variables, a relationship may thus be derived for

either ratio, and such may then be used to define a design feature of the

system. Similarly, other dimensionless parameters may be used to charac-

terize cardiac function and from these, the basis for optimal design of

cardiovascular devices may be derived.

In addition to the utilization of non-dimensional analysis as it

relates to design and experimentation, it has served as a means for making

highly sophisticated mathematical analysis more manageable and applicable.

In a study involving flows in the entrance regions of circular elastic

tubes, Kuchar and Ostrach [8] made use of a non-dimensionalization scheme

in order to obtain an approximate solution satisfying a simplified set of

partial differential equations. Given the complexity of the equ3tions

for fluid velocity distributions, pressure distributions, and motions of

the tube wall, an exact solution becomes impossible to obtain. By

8

introducing the non-dimensionalizing transformations for the variables

involved into the Navier-Stokes equations and simplifying, significant

non-dimensional fluid parameters are obtained as coefficients. These

parameters contain the essential physics of the problem under investiga-

tion and, depending on their relative magnitudes, may be neglected if

shown to be insignificant to the overall problem. In so doing, a simpli-

fied and more manageable set of partial differential equations is obtained

which still contains the basic physics of the problem. The same technique

is applied to the equations governing the motion of the tube wall. A

similar non-dimensionalization scheme was used in the work of Schneck

[16] to examine the phenomenon of pulsatile flow in a diverging circular

channel.

Although the power of the technique ;s unquestionable, it has not

been exploited to its fullest extent to describe the overall function of

the cardiovascular system, particularly with regard to its thermoregula-

tory response. By reviewing the literature of past and present it becomes

evident that the non-dimensional scheme is used as a tool in the simpli-

fication of equations governing specific cardiovascular phenomena only.

It does not appear to be used as a method of modeling the way in which

each aspect of the system interacts with one another and to what extent.

It is this author's desire to explain more fully the integrated function

of the cardiovascular system from the standpoint of heat and mass

transport, and to do so by developing a set of dimensionless parameters

that reveal the essential physics of the system without regard to complex

mathematical analysis or sophisticated experimentation.

CHAPTER III

Method of Analysis

A. The Buckingham Pi Theorem

"A non-dimensional parameter or variable is any quantity, physical

constant, or group of these formed in such a way that all of the units

identically cancel" [16].

Such a statement is the basis for the Buckingham Pi Theorem. Devel-

oped in 1914, it provides a systematic method to develop a finite number

of dimensionless parameters from a'group of variables, where a fixed

number of these variables are repeated in each parameter. In order to

initiate such a scheme, it must first be determined which dimensions are

to be considered fundamental. Most commonly, the choice for these are

mass, length, and time. However, because the problem at hand is complex

and involves variables representing thermal and electrical properties as

well, temperature and charge must be added for a total of five fundamental

dimensions. They are represented by the symbols M, L, T, e, and q,

respectively.

Once this is determined, each variable must be written in terms of

the fundamental dimensions, raised to some appropriate exponential power.

For example, mean arLerial blood velocity would be written as L1T - I ,

while fluid shearing stress would be represented by M1L-1T- 2 . Each of

the total of k (about 600) variables defined in Reference [13] and summa-

9

10

rized in Chapter IV (Table 1) were dimensioned in such a fashion.

The next step in the non-dimensionalization scheme is to select

n variables from among the total. These n variables will form the basis

of the technique in the sense that they will be inherent in all the

final dimensionless parameters. In this case, k is approximately equal

to six hundred and n = 5 (representing the number of fundamental dimen-

sions chosen). The choice for the five starting variables is essentially

arbitrary except that they must contain among them all five fundamental

dimensions. Since this study specifically addressed heat transport pro-

cesses, this author chose variables which reflect thermal concepts related

to the heat transport characteristics of blood. The variables chosen

were: Thermal Conductivity of the Blood (kt): MlLIT-36 -1 , Total Thermal

Capacity of the Blood (C): MIL 2T-2e-1, Specific Heat of the Blood

at Constant Pressure (cp): L2 T 26-1, Convection Coefficient of the

Blood (film coefficient, h): MT-36 "1, and the Faraday Constant (F):

M-1 q1 . These variables are used as "repeating variables" that can be

grouped with any of the remaining variables to determine each new dimop-

sionless parameter.

The essence of the Buckingham Pi Theorem states that, for the above-

chosen variables, an equation may be written which has the form:

kt aCbcp chdFe( )1 = MOLOT0e0qO

The blank indicates where each of the remaining variables is to be inserted.

Rewriting each variable in terms of its own hasic dimensions, the

]1

equation becomes:

(MILlT- 3 8-1 )a . (MIL2T- 20-l)b . (MOL 2T-2-1)c . (MILOT- 3 -l)d

(M-lLOTOOOql)e (_)1 = MOLOTOOOqO

By replacing the blank with each of the remaining k - n variables

individually and by equating corresponding coefficients, five equations

will be obtained that can be solved simultaneously for each of the five

lettered exponents.

For example, in order to create the dimensionless parameter involving

the electrical resistance of the vascular wall, R*, having the dimensions

MIL 2T-160q-2 , one would insert this quantity into the above equation

and solve accordingly:

M: (1 a) + (1. h) + (0 c) + (1 d) + (-1 e) + (1) 0

L: (I * a) + (2 * b) + (2 • c) + (0 d) + (0 e) + (2) = 0

T: (-3 • a) + (-2 * b) + (-2 • c) + (-3 • d) + (0 • e) + (-1) = 0

e: (-1 a) + (-I b) + (-I c) + (-I d) + (0 e) + (0) 0

q: (0 a) + (0 b) + (0 . c) + (0 . d) + (i - e) + (-2) 0

or

a + b +d -e+ = 0

a + 2b + 2c + 2 0

-3a - 2b - 2c - 3d - I = 0

-a -b c - d + 0 = 0

e- 2 =0

12

Solving these simultaneously for the exponents:

a =-4

b= 2

c = -1

d= 3

e 2

Hence, the dimensionless parameter becomes:

kt- 4C 2c p-h 3F2R*

Or:

C2h 3F2R*

kt4Cp

The same technique is performed for each of the k - n remaining

variables to produce k - n unique dimensionless parameters.

13

B. Simplification and Physical Significance of Dimensionless Parameters

Once the dimensionless parameters have been developed, it is conven-

ient to write them in terms of some "standard" dimensionless numbers

that have already been defined for specific groupings of variables. These

numbers (such as the Prandtl Number, Reynolds Number, Peclet Number, and

so on) are common to the fields of solid and fluid mechanics and deal

with such phenomena as magneto-fluid dynamics, heat and mass transfer,

viscous flow, viscoelasticity and diffusion. By isolating combinations

of variables in such a manner, the physical significance of each parameter

becomes more readily realized.

In order to illustrate the technique, examine the dimensionless

parameter involving vascular internal diameter, D: hD, which is recog-kt

nized to be the familiar Nusselt number (Nu). Physically, this represents

a ratio of heat transfer by convection to heat transfer by conduction.

Similarly, the dimensionless parameter involving the asymptotic limiting

value of thixotropic fluid shearing stress under constant load (T.):

CT./(kt 3/cph), represents the ratio of heat generated by thixotropic shear

effects to heat dissipated by a combination of convection and conduction

effects. A physical significance can likewise be assigned to each of

the remaining derived parameters (see Table 1).

Hence, by similar technique, a collection of parameters can he

obtained which shows the types of phenomena that are prevalent to cardio-

vascular function. The results of this investigation are presented in

Chapter IV and discussed in the sections that follow.

CHAPTER IV

Tabulated Results of the Analysis

Table I presents the results of a non-dimensional analysis of

cardiovascular function and thermoregulation. It lists each variable,

together with its respective symbol, dimensions, derived dimensionless

parameter, and the physical significance of this parameter. The vari-

ables are grouped according to whether they are related to the blood,

the heart, the vascular system, or to some overall aspect of cardiovas-

cular function and thermoregulation.

By applying the Buckingham Pi Theorem to each variable, a unique

dimensionless parameter was derived involving that variable and a combi-

nation of other thermal and electrical variables. Each derived parameter

has a unique physical significance (column 5 of Table 1) that reflects

the role of the cardiovascular system in thermoregulation. Through the

development of these parameters, thermal, electrical, and mechanical

variables are coupled together in meaningful groupings that provide a

powerful insight into heat transport phenomena, without having to get

involved in tedious mathematical formulations. Depending on the magni-

tude of the individual variables (a subject which was not specifically

addressed in this work) comprising the dimensionless groupings, each

parameter reflects the significance and relative importance of various

electrical and thermal processes involved in thermoregulation. Derived

dimensionless parameters and their corresponding physical significance

are discussed in Chapter V in terms of their relationship to the blood,

14

15

heart, vascular system and overall thermoregulatory response.

In the process of non-dimensionalizing and interpreting the resulting

parameters, a series of fundamental scales associated with electrothermo-

dynamic effects was derived naturally. These results are presented in

Table 2 and include scales for mass, length, time, temperature, and

charge. For each of these physical quantities, a unique scale has been

derived that involves the variables originally chosen as the foundation

for the Buckingham Pi method of non-dimensionalization. The combinations

of these variables, as shown in Table 2, represent fundamental scales

for the various electrical and thermal processes that are associated

with the regulation of body temperature. This is significant in that it

provides the ability to derive a unique dimensionless parameter for any

variable by simply knowing its basic dimensions. For example, the Mass

Density of Blood, Pf, has the dimensions ML- 3. This is rewritten in

terms of the derived scales:

(C/cp)(kt/h)-3 = Ch 3 /c pkt 3

Hence, a unique dimensionless parameter is created for this variable by

forming the ratio of pf to this quantity: pf/(Ch3/Cpkt3). This ensures

a non-dimensional parameter and precludes a longer and more formal anal-

ysis that is required using the Buckingham Pi Theorem. This not only

serves to simplify the task of formulating the parameter itself, but it

also facilitates the analysis of what the parameter represents from a

physical standpoint.

Further expanding on the concept of scales associated with electro-

thermodynamic processes, A series of kinematic, kinetic, thermal, and

16

transport quantities were written in terms of the fundamentally derived

scales. These results are presented in Table 3. Having these quantities

in this form simplifies the task of interpreting each derived parameter.

The physical significance of many of the dimensionless parameters became

evident after identifying groupings of these quantities from within the

parameter itself. In this way, the translation between mathematical

entity and physical phenomenon was realized.

The results of this analysis provide a valuable way to characterize

cardiovascular thermoregulation through the development of a tangible

set of numerical parameters consisting of several thermal and electrical

variables that were selected for the purpose of the non-dimensionaliza-

tion scheme. The resulting parameters may be further written in terms

of more traditional dimensionless numbers, such as those listed in Table

4. These dimensionless numbers are common to the fields of solid and

fluid mechanics and incorporate various heat and mass transport phenomena.

Several of the dimensionless parameters derived in this study were writ-

ten in terms of the traditionally defined numbers. These results are

presented in Table 5. Here, variables having the same basic dimensions

are grouped together for purposes of reference. A sampling of some

variables, along with the resulting combination of dimensionless numbers,

is provided to illustrate that the parameters derived from this study

relate not only to heat and mass transport within the cardiovascular

system, but they reflect even more fundamental concepts of fluid flow

and heat transfer inherent within any dynamic system. Since some of the

variables defined in Reference [13] are inherently dimensionless already,

17

there was no need to involve these variables in any further non-dimen-

sional analysis. They should, however, be recognized as important

controlling factors in the system's attempt to protect itself from tem-

perature extremes and are indeed parameters in and of themselves. A

listing of these dimensionless variables is presented in Table 6.

Moreover, other variables defined in Reference [13] are difficult to

quantify in the non-dimensional sense, but, either from a chemical or

physical standpoint, they enhance or impede the thermoregulatory capacity

of the cardiovascular system. These variables are found in Table 7 and

relate to characteristics of the individual, climate, clothing, concen-

tration and properties of carrier molecules in the blood, diet of the

individual, predilection to cold stress, properties of the myocardial

muscle, statistical parameters, and physical properties of the vascular

system. A combination of some or all of these variables may interact to

control the manner and the efficiency with which the system protects

itself from an assaulting environment. To complete the list of variables,

consideration must be given to the various enzymes, hormones, nutrients,

vitamins, buffering ions, minerals and electrolytes, trace elements,

blood gases, and byproducts of metabolism (Tables 8 - 12) that are found

within the body and whose concentrations directly influence the processes

of heat and mass transport.

CHAPTER V

Discussion of Results

The purpose of this work was to derive the dimensionless parameters,

themselves. To actually calculate them, as well, would have been too

formidable a task, and so this is left for a future study. However,

some general qualitative discussion of the results obtained is in order

at this point. For example, through the derivation of dimensionless

parameters involving blood-related variables, various heat transfer pro-

cesses dre revealed that take place due to the inherent properties of

blood itself. An examination of each derived parameter and its corre-

sponding significance shows that blood has the ability to store heat,

depending on its total thermal capacity, and to distribute this thermal

energy throughout the body via momentum transport, molecular dispersion,

ordinary diffusion, electromotive diffusion, and viscous dissipation.

Each of these processes is incorporated in a respective associated Pi

parameter. The variables involved are mathenatically coupled with the

established set of thermal and electrical variables chosen for the non-

dimensionalization scheme, i.e., Thermal Conductivity of the Blood (kt),

Total Thermal Capacity of the Blood (C), Specific Heat of the Blood at

Constant Pressure (cp), Convection Coefficient of the Blood (film co-

efficient, h), and the Faraday Constant (F).

If one examines the derived non-dimensional parameter involving the

Dynamic Elastic Modulus of Blood (Ed(t)/(kt/cp)) it is possible to

interpret this ratio as the energy per volumetric flow rate of blood

18

19

associated with elastic effects, to the corresponding thermal energy

(conduction) per volumetric flow rate of blood. This represents a ratio

of well-defined physical quantities that relate the ability of the cardio-

vascular system to transport heat by each of these physical mechanisms.

Similarly, the parameter involving the Dynamic Viscous Modulus of Blood

(Ej(t)/(kt/cp)) represents the ratio of heat per volumetric flow rate

associated with viscous dissipation in blood, to that transported by

conduction. Other parameters involve the ratio of energy transported by

molecular dispersion (Eddy Diffusion Coefficient, DE), electromotive

diffusion (Electromotive Diffusion Coefficient, Ds*), and ordinary dif-

fusion (Ordinary Mass Diffusion Coefficient, Ds) to heat transported

by mass and mass flux through electrothermodynamic processes. In addition

to these parameters involving energy and heat transport, a ratio of

heat generated per mass flow rate of blood due to viscous dissipation

and heat transported per mass flow rate of blood through electrothermo-

dynamic processes is revealed in the derived parameter involving the

Kinematic Viscosity of Blood (vf/(kt4 /Ch3 )).

Volume ratios were also derived using other variables. The parameter

involving Total Blood Volume (VB/(kt 3/h3 ) represents the ratio of total

blood volume to a characteristic volume of blood associated with thermo-

dynamic events. Similarly, other non-dimensional volume ratios were

defined for variables associated with other aspects of cardiovascular

thermoregulation, such as End Diastolic Volume (EDV/(kt 3/h3)), End Sys-

tolic Volume (ESV/(kt 3/h3 )), and Stroke Volume (S.V./(kt 3/h3)). Each

of these parameters involves the ratio of the specific heart volume

20

involved to the volume of blood characteristic of thermodynamic events.

Each variable in Table I with dimensions of time creates a unique

parameter that addresses the ratio of the respective time scale associ-

ated with that particular event to the time scale associated with various

thermodynamic processes. These parameters involve such time scales as

the Time Spent in Core (Blood) (tc/(Ch/kt2 )) and the various ECG inter-

vals, such as the Q-R-S Complex Interval (tQRS/(Ch/kt 2 )). Parameters

that involve frequency are merely reciprocals of time scales associated

with the related phenomena. For example, the derived dimensionless

parameter involving the Natural Frequency of the Vascular Wall

(wn/(kt2 /Ch)) represents the ratio of the time scale associated

with diffusion of heat through blood to the time scale associated with

the kinematic translation of the vascular wall.

Variables involving pressure and stress are represented as ratios of

certain energies per unit volume. The derived dimensionless parameter

for Osmotic Pressure (np/(kt 3 /Ccph)), for example, represents the ratio

of potential energy per unit volume associated with molecular kinetic

energy driving mass across the membrane as a result of the existence of

concentration gradients, to potential energy per unit volume associated

with mass transport resulting from thermal gradients.

The examples presented above are not meant to be exhaustive. They

merplv serve to illustrate how one can interpret the physical significance

represented by several types of non-dimensional parameters. Variables

associated with the heart produce dimensionless parameters that reflect

the role of the heart in thermoregulation. Such parameters provide a

21

relationship between the function of the heart-- characterized by various

ECG intervals, pressures, cardiac output, cardiac volumes, heart rate;

and so on-- and the electrical and thermal processes that take place

within the body to control and regulate temperature. Similarly, variables

related to the vascular system are found in parameters that reflect the

role of the vascular system in thermal regulation. Energy per unit volume

associated with elastic storage in vascular tissue, heat per unit volume

associated with viscous dissipation in vascular tissue, energy transported

across membranes by mass flux due to ordinary diffusion, electromotive

diffusion, electromagnetic diffusion, and osmosis, and energy per unit

volume associated with stress in the vascular wall are some of the quan-

tities incorporated in the dimensionless parameters involving the vascular

system.

By applying the techniques of non-dimensionalization to each varia-

ble, a dimensionless parameter is derived that relates this variable to

the overall process of thermoregulation by creating a unique ratio of

identifiable events. The resulting parameter provides a physical quanti-

ty to which a numerical value may be assigned. Depending on the value

thus obtained, one may look, then, for a significant correlation between

the function of the cardiovascular system under consideration and the

mechanisms for controlling temperature within the body.

By examining the values of the derived fundamental reference scales

found in Table 2 (as calculated in Reference [17]), valuable information

is provided in interpreting the relative significance of various heat and

mass transfer events taking place within the cardiovascular system. For

22

example, assuming a normal cardiac rate of 80 cycles per minute, the time

scale for the transport of unsteady inertial phenomena is calculated as

follows: 1/ = 1/2Tf = 1/27r(80) = 0.002 minutes, or, about

0.12 seconds. Similarly, calculating the time scale associated with the

heat transport characteristics of blood, Ch/kt 2, (based on values provided

in Reference [17]) one obtains the following result: Ch/kt 2 =(4.0

Kcalories/°C) (174.96 KCal/C-hr-m2 )/(0.48 KCal/°C-hr-m)2 = 2,975.2 hours,

or about four months! The results of this simple calculation

reveal that the events associated with the transport of blood through the

cardiovascular system occur nearly 100 million times faster than do those

associated with the ability of the fluid to dissipate heat through its

own thermal properties[17]. Therefore, blood, being a poor conductor of

heat, dissipates the heat generated within the body most effectively by

physically carrying that heat through blood flow rather than waiting for

the heat to dissipate by itself through the fluid - a process that would

certainly lead to death since the body would not be able to handle the

heat at the rate that it would be generated. Hence, blood can rapidly

absorb and transport large quantities of heat, but it cannot easily allow

that heat to move within the fluid as a result of thermal gradients

associated with conductive and convective processes.

By also examining the calculated value for the reference temperature

scale, 1.04 x 10-22oC or nearly 00C, one recognizes this as the "standard"

thermodynamic reference temperature for the measurement of most heat

transfer processes. The analysis thus suggests a reference temperature

scale that is more generally associated with standard conditions (00C and

23

atmospheric pressure) than with physiologic conditions (37°C 3nd atmos-

pheric pressure). Similarly, the derived length scale, kt/h, has a

calculated value of 2.75 millimeters. This is on the order of magnitude of

the length of a capillary, the mean radius of a medium sized artery (or

vein), or the average wall thickness of any major blood vessel, such as

the aorta or the vena cava. The naturally derived mass scale, C/cp =

4.35 kilograms, represents a mass scale associated with the heat transport

characteristics of the fluid and is defined to be the total blood mass

divided by the specific heat ratio of the fluid.

Therefore, as a result of this analysis, various fundamental scales

associated with electrothermodynamic processes taking place within the

cardiovascular system were identified and calculated. Through this

process, the response of the cardiovascular system to changes in the body

temperature is better understood and is able to be quantified. Such

quantities may, in turn, be applied to each derived parameter to charac-

terize its importance to overall cardiovascular thermoregulation.

Since the cardiovascular system is extremely complex, as is evident

by the extensive list of variables presented in Tables 1-12, many

experimental limitations are imposed upon any comprehensive study that

might be undertaken. Consequently, the need arises for a more fundamental

approach to determine a working set of parameters that characterize the

system and, from these, to select which factors would be most significant

in future modeling and experimental studies. The power of a non-dimen-

sionalization scheme provides a method for accomplishing this task.

By isolating the thermal, mechanical, and chemical variables associated

24

with the cardiovascular system and by developing from these a set of

non-dimensional parameters, a framework is established for modeling the

heat and mass transport response of the system to a given thermal input.

This analysis has served to develop these working parameters and to iden-

tify the physical quantities that predict and control such a response.

CHAPTER VI

Conclusion

As a result of the non-dimensional analysis described in the prece-

ding chapters, several significant concepts regarding cardiovascular

function and thermoregulation have been developed. Using the technique

of the Buckingham Pi Theorem, a dimensionless parameter was derived for

each isolated variable associated with heat and mass transfer within the

cardiovascular system. This parameter is unique to this variable and

directly couples the related phenomenon to other thermal, mechanical, and

electrical events that take place within the system.

Through the process of non-dimensionalization, a series of fundamen-

tal scales for each of the five basic dimensions was derived naturally

from the resulting parameters. These were identified and quantified in

Table 2. Although they may appear to be quite simplistic at first glance,

they contain powerful information that permits the derivation of any

non-dimensional parameter directly, without having to employ rigorous

calculations necessitated by the Buckingham Pi Theorem. The dimensions

of any selected variable can be written in terms of the derived fundamental

scales. By creating a ratio of the variable to the resulting combination

of fundamental scales, the non-dimensional parameter is immediately formed.

The ability to create dimensionless parameters in this way is highly

significant. The technique ensures that the resulting parameter is non-

dimensional, and it can be performed immediately, without solving any

equations beforehand. It must be reminded, however, that the simplified

25

26

method of non-dimensionalization is possible only because the fundamental

scales were initially derived for the problem - a process that required

the structure and methodology afforded by the Buckingham Pi Theorem.

By examining the quantitative values for the fundamental scales that

resulted from this analysis (Table 2), significant information is obtained

regarding heat and mass transfer within the cardiovascular system. The

value obtained for the fundamental time scale, 4.16 months, appears vast

when compared with the time scale for the transport of unsteady inertial

phenomena - calculated to be about 0.12 seconds. This indicates that the

events associated with the physical transport of blood through the

cardiovascular system occur nearly 100 million times faster than do those

associated with the ability of the fluid to dissipate heat through its

own thermal properties. Blood is shown to be a poor conductor of heat.

Hence, the human organism could not survive if it depended on blood,

alone, to disperse heat due to the thermal properties of its constituents.

There must be another mechanism to enhance heat transport and dissipation,

and this occurs through the physical movement of the blood, or circulation.

The value obtained for the fundamental time scale emphasizes that conduc-

tive heat transfer in the cardiovascular system is much smaller than

that due to bulk flow.

Similarly, the other derived fundamental scales each bring a physical

significance to the analysis through their respective numerical values.

More important than the actual values calculated for these scales is the

comparison between these values and the values associated with other

events occurring within the cardiovascular system. Through such a compar-

27

ison, the relative significance of each event to overall cardiovascular

thermoregulation is established. This type of information must be

known before a more complicated analysis is attempted and, in fact, may

help to develop fundamental assumptions (and simplifications) of the

behavior of the cardiovascular system in its response to a thermal stress.

A non-dimensional analysis was chosen to approach this problem for

the following reasons: First, the technique exploits a very powerful

means for grasping the fundamental features and essential physics of any

given problem before attempting to pursue a more complicated mathematical

formulation. Second, it allows one to identify explicitly the important

parameters that need to be measured when performing experiments that deal

with physiological responses to environmental stresses. (The parameters

are developed through a systematic technique and many parameters, that

otherwise could only be identified through complex differential equations,

are derived with relative ease using a non-dimensional scheme.) Third,

criteria are established that can be used to screen individuals for their

susceptibility to thermal (hot or cold) stress. Fourth, the resulting

non-dimensional parameters may be used as design parameters for the

development of physiologic training maneuvers and acclimatization exerci-

ses, as well as for the fabrication of thermal protection clothing and

gear. And fifth, the parameters allow one to distinguish between first-

order effects and higher-order effects in the analysis of data obtained

from well designed thermal stress experiments.

Given the complexity of the cardiovascular system, it is necessary

to develop a framework for future studies by identifying what parameters

28

are most significant to the analysis. Otherwise, substantial time and

wasted energy may be spent to produce results that are insignificant and

unrelated to the original problem. To whatever extent non-dimensional

techniques may contribute to directing future research, they should be

pursued in a continuing effort to improve the health, comfort, and

understanding of man.

REFERENCES

1. Barenblatt, G.I., Similarity, Self-Similarity, and IntermediateAsymptotics, New York, Consultants Bureau, Chapter 1, pp. 13-30, 1979.

2. Bridgman, P.W., Dimensional Analysis, New Haven, Connecticut, YaleUniversity Press, 1931.

3. Cooney, D.O., Biomedical Engineering Principles, New York, MarcelDekker, Inc., Chapter 5, pp. 93-155, 1976.

4. Eshbach, O.W.(Editor), Handbook of Engineering Fundamentals, NewYork, John Wiley and Sons, Inc., Section 3, pp. 3-01 - 3-45,1936.

5. Holman, J.P., Heat Transfer, New York, McGraw-Hill Book Company,Third Edition, 1972.

6. Huntley, H.E., Dimensional Analysis, New York, Dover Publications,Inc., 1967.

7. Hwang, N.H.C., Gross, D.R., and Oatel, D.J., Quantitative Cardiovas-cular Studies: Clinical and Research Applications of EngineeringPrinciples, Baltimore, MaryTand, University Park ress, pp.

4!51, 1979.

8. Kuchar, N.R., and Ostrach, S., Flows in the Entrance Regions ofCircular Elastic Tubes, Case Western Reserve University, Schoolof Engineering, Division of Fluid, Thermal and AerospaceSciences, FTAS/TR-65-3, June 1965.

9. Land, N.S., A Compilation of Nondimensional Numbers, Washington, D.C.,National Aeronautics and Space Administration, 1972.

10. Li, J. K-J., Fich, S., and Welkowitz, W., "Similarity Analysis ofMammalian Hemodynamics," in Welkowitz, W. (Editor),Proceedings of the Ninth Northeast Conference on Bioengineering,ew York, Pergamon Press, pp. 50--53,181

11. McComis, W., Charm, S., and Kurland, G., "Dimensional Analysis ofPulsating Flow - - An Introduction," in Copley, A.L. (Editor),Proceedings of the Fourth International Congress on Rheology,New York, Interscience Publishers, pp. 231-242, 1965.

12. Schepartz, B., Dimensional Analysis in the Biomedical Sciences,Springfield, Illinois, Charles C. Thomas, 1980.

13. Schneck, D.J., A Non-Dimensional Analysis of Cardiovascular Response

29

30

To Cold Stress: Part 1, Identification of Physical Param-eters ht Govern he Thermoregulatory Function of theCardiovascular System, Naval Medical Research InstituteTechnical Information Service, Technical Report Number NMRI83-51, NTIS Accession Number AD-A138 710/9, September, 1983.

14. Schneck, D. J., Biomechanics of Striated Skeletal Muscle, SantaBarbara, California, Kinko's Professor Publishing Service, 1986.

15. Schneck, D.J., Principles of Mass Transport Across BiologicalMembranes, Virginia Polytechnic Institute and State University,College of Engineering, Department of Engineering Science andMechanics, Technical Report Number VPI-E-83-07, March 1983.

16. Schneck, D. J., "Pulsatile Blood Flow in a Diverging CircularChannel," Ph.D. Dissertation, Cleveland, Ohio, Case WesternReserve University, 1973.

17. Schneck, D.J., and Starowicz, S., "A Non-Dimensional Approach tothe Analysis of the Heat-Transport Characteristics of Whole HumanBlood," Paper submitted for publication, 1986.

18. Weast, R.C. (Editor), CRC Handbook of Chemistry and Physics, 52ndedition, Cleveland, Ohio, The Chemical Rubber Company, pp. F-266-F-283, 1971.

Appendix

31

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TABLE 2

DERIVED FUNDAMENTAL SCALES ASSOCIATED WITH ELECTROTHERMODYNAMIC EFFECTS

DIMENSION SCALE VALUE([17])

Mass M C 4.35 Kilograms

Cp

Length L kt 2.75 Millimeters

h

Time T Ch 4.16 Months

kt2

Temperature 0 kt6 1.04 x 10- 22 0C,or Nearly OCC2cpi, q

Charge q CF 4.2 x 13' Coulombs

p

86

TABLE 3

OTHER DERIVED SCALES ASSOCIATED WITH ELECTROTHERMODYNAMIC EFFECTS

KINEMATIC QUANTITIES

Displacement (L): kt/h

Velocity (L/T): kt3/Ch2

Acceleration (L/T2 ): kt5/C2h3

KINETIC QUANTITIES

Angular Momentum (r x mv) kt4/cph3

Angular Impulse (r x FT)j

Force (F): kt5/Ccph 3

Moment (r x F): kt6/Ccph4

Momentum (Mv) kt3cph 2

Impulse (FT) I

Power (FL/T): kt8/C2cphS

Pressure or Stress (F/L2 ): kt3/Ccph

Work (FL) 1 ktG/Ccph 4

Energy (FL)J

THERMAL OR THERMODYNAMIC QUANTITIES

Heat Capacity (FL/e): kt6/Ccph 4

Temperature (6): kt6/C2cph4

TRANSPORT QUANTITIES

Diffusion (L2 /T): kt"/Ch3

Volumetric Flow Rate (L3/T): kt5/Ch4

87

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88

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TABLE 6

LIST OF VARIABLES WHICH ARE ALREADY

DIMENSIONLESS BY DEFINITION

DIMENSIONLESS VARIABLE SYMBOL

Blood

Arrangement Factor

Dielectric Constant for Blood

Emissivity of Blood e

Hematocrit (%) H

Non-Newtonian Index n

Percent 02 Saturation (Hemoglobin) % 02

pH pH

Species - Specific Constants for RheologicEquation for Blood C1 , C2 , C3, C4

Specific Heat Ratio for Blood y

Heart

Efficiency of Cardiac Cycle (Overall) nt

Efficiency of Cardiac Muscle Contraction nm

Efficiency of Energy Conversion from StrokePower to Fluid Kinetic Energy nk

Efficiency of Energy Conversion into Stroke Volume ns

99

TABLE 6 (continued)

DIMENSIONLESS VARIABLE SYMBOL

Efficiency of the Heart nh

"Other" Efficiencies in Cardiac Cycle nn

Frequency Component of ECG Spectrum ni

Korotkoff Heart Sounds (db)1 , (db)2 , (db) 3

(db)4

"Other" Heart Sounds (db)i

Phase Lead Angle for Oscillating Blood Flow f

Magnitude of f I f

Probability Density Function p(X)

Probability Distribution Function P(X)

Probability Range Function f p(X)dX

Vascular System

Amplitude Magnification Factor (for cyclicloading of vascular wall) Q

Branching Angles of Vascular Geometry 0

Coefficient of Friction (Wall Pore and Pore Fluid) sf

Coefficient of Friction (Pore Fluid and Solute) Sp

Coefficient of Friction (Wall Pore and Solute) ss

Constant Strain in Vascular Wall co

Dielectric Constant for Vascular Wall K*

Elastic After-Effect (Strain) E*(t)

Emissivity of Vascular Wall Ew

Form Factor m

100

TABLE 6 (continued)

DIMENSIONLESS VARIABLE SYMBOL

Fraction of Vascular Area Available for HeatTransfer to Body Core Ac

Fraction of Vascular Area Available for Heat

Transfer to Environment (near skin surface) Ae

Friction Factor fR

Generalized Poisson Ratios for Vascular 0iSmooth Muscle i = 0, 1, N

Maximum Strain in Vascular Wall Emax

Minimum Strain in Vascular Wall Emin

Number of Anastomoses Open at any Time N2

Number of Collateral Pathways at any Time N3

Number of Patent Blood Vessels at any Time NI

Number of Pores/Membrane Area c

Percent by Weight:

Collagen Wc

Elast:., We

Smooth Muscle Tissue Ws

Phase Angle for Viscoelastic Blood Vessel Wall w

Magnitude of w I wl

Pore - Free Surface Area (% of Av) Ao

Porous Surface Area (% of Av) Ap

Reflection Coefficient (Staverman Factor) b

Roughness Factor /D

101

TABLE 6 (continued)

DIMENSIONLESS VARIABLE SYMBOL

Specific Heat Ratio at Vascular Wall y*

i = O,1, N

Stimulation Coefficients for Vascular Smooth Muscle qi

Time-Dependent Strain in Vascular Wall &(t)

Tortuosity Factor for Pores 6*

Total Number of Cycles (required for hysteresisto reach steady-state) M

Vascular Smooth Muscle Strain &m( m

Cm 3,

Volume Fraction of Water in Pores p

Miscellaneous

"Activity Factor"

Ambient Humidity Ratio (%) HR

Degree of Nonlinearity for any Model N

Dissociation Constant for Electrolytes Di

e = 2.718 e

Ionic Valence of Species S Z

Partition Coefficient Y)

Pi - 3.14159 (circumference/diameter)

Respiration Quotient RQ

Solubility /1'

Solubility Coefficient

102

TABLE 7

OTHER VARIABLES THAT CAN BE ASSOCIATED WITH CARDIOVASCULAR

FUNCTION AND THERMOREGULATION

Characteristics of Individual

Age Yr

Anthropometric Build

Height Ht

Muscle Mass

Physical Condition

Cardiovascular System

Central/Sympathetic Nervous System

Endocrine System

Musculoskeletal System (exercise)

Weight Wt

Climate

Season of the Year (Fall, Winter, Spring, Summer)

Relative Humidity

Amount of Daylight

Presence or Absence of Wind (Wind Velocity) U

Ambient Temperature TA

Time of Exposure te

Clothinq

Amount

103

TABLE 7 (continued)

Type

Thermal Properties

Concentration and Properties of Carrier Molecules in Blood

Affinity for Substrate (Stereospecificity)

Deformability

Degree of Hydration

Degree of Saturation

Geometry (Stereochemistry)

Mobility

Diet of Individual

Chemical Composition of Blood

Alcohol Consumption

Quantity and Type of Food Intake (especially fats andcarbohydrates)

History

Family History

Predilection to Cold Stress

Frequency of Previous Exposure

Time Between Exposures

Myocardial Muscle

"Contractility"

104

TABLE 7 (continued)

Statistical Parameters

General Wave Shape of ECG

Kurtosis (Flatness Factor)

Skewness

Width Factor

Vascular System

Cold Induced Vaso-Dilatation CIVD

Geometry of Vascular Wall

Branches/Bifurcations

Convergence/Divergence

Curvature

FPrallelism/Taper

Geometry of Vascular Smooth Muscle

Length of Blood Vessels

Development of Flow (inlet length)

Effective Length of Pressure Drop

End Effects

Pulse Wave Reflection

Orientation and Distribution of Blood Vessels

Arterio-Venous Anastomoses AVAs

Pores

Cross Sectional Area

Cross Sectional Contour

Degree of Patency

105

TABLE 7 (continued)

Length

Mean Diameter

Number

Properties of Material Within Pore

Smoothness

Spatial Orientation

Symmetry

Wall Configuration

Symmetry of Vascular Cross Section

Vasoconstriction

Arteries

Capillaries

Veins

Vascular Lumen Patency

Degree of Extravascular Compression

Degree of Internal Occlusion

Level of Smooth Muscle Tone

Vascular Tone

Venous Shunt Response VSR

Atrio-Ventricular Countercurrent Heat Exchange(see N1 , N2 , N3 )

106

TABLE 8

ENZYMES

Dehydrogenases:

glucose-6-ca-hydroxybutynic dehydrogenase HBD

glutami c dehydrogenase GDH

isocitnic dehydrogenase I CD

lactic dehydrogenase LOH

maleic dehydrogenase MDH

Transaminases:

glutamic oxaloacetic transaminase GOT (SGOT)*

glutamic pyruvic transaminase SGPT**S=serum)

acid phosphatase

aldolase

alkaline phosphatase

amino peptidase

amyl ase

arachidonic acid (test for presence of enzymes)

aromatic L-amino acid decarhoxylase

chol inesterase

creatinine phosphokinase CPK (-MM, -MB, -BB)

dopami ne-beta-hydroxyl ase

gu anas e

kinase I

kinase 11 (precursor- kininogen)

107

TABLE 8 (continued)

k ini nase

1lipase LPP

oi-nithine carbamyl transferase OCT

phenyl ethanol ami ne-N-methyl transferase

prostacyci in

reni n

tyrosine hydroxylase

108

TABLE 9

HORMONES

Catecholamines (Blood and Vascular Smooth Muscle Tissue):

epinephrine (adrenalin)

norepinephrine (noradrenalin) (precursor- dopamine)

local myocardial catecholamines

Steroids:

aldosterone

corticosterones "A", "B", or "S"

cortisone

deoxycorticosterone

hydroxycortisone (cortisol)

acetylcholine Ach

adrenocorticotropic hormone ACTH

angiotensin I and II

antidiuretic hormone (vasopressin) ADH

calcitonin

central nervous system chemical releasing factors

corticotrophin releasing factor CRF

dopa

glucagon

growth hormone release inhibiting hormone (somatostatin) GHRIH

growth hormone releasing factor GHRF

histamine

109

TABLE 9 (continued)

insulin

kinins:

bradyki ni n

kallikrein

kallidin (lysin bradykinin)

parathormone

prostaglIand ins D2, E2, F2

serotonin

somatotrophin (growth hormone) GH

thyrocal citonin

thyrotrophi n TSH

thyrotrophin releasing hormone TR H

thyroxi n

tri iodothyronine

110

TABLE 10

NUTRIENTS

water H20

Amino Acids:

tyrosine

tryptophan

Proteins:

total protein concentration [P]

albumins [A]

total protein minus albumin [TPMA]

fibrinogen [F]

globulins: [G]

a ceruloplasmin (al, a2 , 6, y)

immunoglobulins IgA, IgG, IgM

vascular wall:

elastin [El

collagen [C]

Carbohydrates:

glucose

mucopolysaccharide in vascular wall [M]

Lipids and Fatty Acids:

111

TABLE 10 (continued)

cholesterol (fats): (vitamin F)

free

esteri fied

total

triglycerides

phosphol ipids

total lipids

subcutaneous fat content (insulation) [fat]

effective insulation thickness ofhuman body IT

112

TABLE 11

* VITAMINS

A retinol

B complex:

BI thiamine

B2 riboflavin (component- biotin)

B3 niacin/nicotinamide

B4 adenine

B5 pantothenic acid

B6 pyridoxine

812cholinefolic acidinositolpara-amino benzoic acid PABA

C ascorbic acid

C complexacerolahesperidinrutin

D calciferol

E tocopherol a-, 6-

E complex

K (clotting)

113

TABLE 12

BUFFERING IONS

ammonium NH4 +

bicarbonate HCO 3-

hydrogen (pH) H+

phosphate P04 -

3

sulfate S04 -2

MINERALS AND ELECTROLYTES

calcium Ca

chloride CI

copper Cu

iron Fe

magnesium Mg

phosphorus P

potassium K

sodium Na

TRACE ELEMENTS

cadmium Cd

chromium Cr

cobalt Cb

fluorine F

iodine I

11*4

TABLE 12 (continued)4

protein-bound iodine PBI

lithium Li

manganese Mn

molybdenum Mo

selenium Se

sulfur S

zinc Zn

BLOOD GASES

carbon dioxide CO2

oxygen 02

arterio-venous oxygen concentration difference A[02]I

hemoglobin (percent saturated) Hb

BYPRODUCTS OF METABOLISM

bilirubin

blood urea nitrogen BUN

carbon dioxide CO2

creatine

a creatinine

lactic acid

water

115