BASIS FOR A CIRCULATORY MODEL. Book Final/Chapter 3.pdf · BASIS FOR A CIRCULATORY MODEL. At its simplest the heart and circulation cons titute a stirring system which, by providing

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CHAPTER.3.

BASIS FOR A CIRCULATORY MODEL.

At its simplest the heart and circulation constitute a stirring system which, by providing oxygen andnutrients on the one hand, and removing metabolites and waste products on the other, maintains theconstancy of the internal environment within narrow limits. It is the efficiency with which it doesthis which we are about to explore in mechanical terms.

In physical terms, living organisms represent sophisticated systems for assimilation, production,and/or distribution and regulation of energy; either between different regions of the particular lifeform, or between the organism and its environment. Unless the available energy is suitablyregulated, the mechanism rapidly becomes disorganised, and the system unsustainable. Becausecomplex organsims are based on watery solutions (including colloidal solutions) of organic andinorganic molecules, regulation and distribution of energy largely depends on the physical propertiesof this water based structure. When the energy available to the fluid is increased, it appears asincreased motion of the constituent particles which make up the solution. The motion can berandom and disorganised resulting in increased heat content, and rising temperature, or it maybecome more organised, with movement of particles in a particular direction, with each potentiatingthe motion of its neighbours, and resulting in overall movement, in whole or in part, of the body offluid .The resulting movement may be recognised and measured as momentum, representing thevolume of fluid moved, and the linear velocity it is given. The mechanisms which regulate fluid flowwith respect to heat production, are those which determine circulatory activity.

The feature which should be emphasised, concerns the method of initiation of the circulatoryresponse, which will not occur unless certain conditions associated with cellular function arefulfilled. Activity of the cells produces venous return by giving linear velocity to the quantity offluid exchanged with the cells. The resulting momentum becomes the initiating factor for circulatoryactivity, and the heart responds in an appropriate fashion when blood with a particular momentumis presented to its chambers.

In any consideration of circulatory function, first priority has to be given to the activity of theventricle in expelling blood into the aorta at each beat . The amount of blood thus expelled i.e. thestroke volume, must be related not only to the venous return, but also to the requirements of thetissues for oxygen and/or nutrients, and for the removal of metabolites both solid and gaseous. Ourunderstanding of this function stems primarily from the observations of three great innovators.

William Harvey in 1628 first put forward a comprehensive account of the circulation of the blood,despite the fact that circulation through the capillaries joining the arteries to the veins was at thattime incapable of direct observation. His views have subsequently been abundantly confirmed.

Claude Bernard in his "Introduction to the Study of Experimental Medicine" published in 1865,described the idea of the `internal environment'. By systematic observation he arrived at theconclusion that a function of the circulation is to maintain the constancy of composition of the fluid

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surrounding individual body cells. This view is basic to modern physiology and therapeutics.

The conclusion by Ernest Starling early last century, that the capacity of mammalian heart musclefor performing external work is directly related to the initial length of the cardiac muscle fibres, aconclusion already inherent in the work of Frank on frog heart muscle, and parallelled by Blix, andalso by A.V.Hill and his associates, in work on skeletal muscle, has influenced all majorinvestigators into cardiac function since his time. The present account is no exception.

The mammalian circulation transfers fluid volume and energy between cells and organs of the bodyin a closely regulated way, to maintain the distribution of function in a continuing fashion for thelifetime of the individual. It is the close regulation of kinetic energy associated with fluid volumewhich is the necessary criterion for the persistence of the living state, and life ceases if thismovement is interrupted for a significant period. The volume of fluid moved, and the linearvelocity of fluid motion are the essential features in assessing the energy involved.

Regulation of circulatory activity requires maintenance of the concentrations of the respiratorygases, carbon dioxide and oxygen, and the effect of lactate concentration on energy transfer acrosscell membranes also alters the distribution of energy between the cell and the extra-cellular fluid,and between the amount of energy available for external work of the cell, compared with thatprovided for tissue perfusion. The relevant relationships are between circulatory length and theproduct of cell concentrations of oxygen and carbon dioxide; and between the square of circulatorylength and the product of the concentrations of lactate and carbon dioxide prevailing in the cells ofthe effector tissues responsible for external cell work (e.g., muscle contraction). An algebraicmodel based on these relationships may be constructed where the necessary parameters are allfunctions of the gas concentrations maintained in the active 'effector' cells. Diagrams setting outthese relationships constitute figures 1 to 4, which are provided with descriptive text to convey theprinciples underlying the model.

Contraction of the ventricle imparts momentum to the volume of blood contained in that particularcirculation (systemic or pulmonary) and the energy produced by ventricular contraction has twoessential mechanical effects. Part of the energy is used to overcome the resistance to flow presentedby the anatomical structure of the blood vessels, together with the resistance offered by the changingviscosity of the blood. The total resistance to flow may be expressed as a ̀ momentum equivalent'.

The second requirement is to ensure that the blood in the circulation retains sufficient of the addedmomentum to enable blood to be returned to the heart, and fill the ventricles adequately before thenext contraction.

These ideas seem to the present day observer, to be implicit in Harvey's view of the circulation,which he based largely on anatomical considerations. Indeed, the idea of `momentum' at the timeof Harvey's original publication (which appeared 14 years before the birth of Newton in 1642, and58 years before the publication of the ̀ Principia') remained largely unformulated, and the conceptionof the circulation of blood at that time, was a most daring and original one. For without some wayof estimating the quantity of motion given to the advancing blood, further progress in understanding

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the manner in which that momentum was utilised or dissipated within the body, would not be readilyachievable.

By the time Newton's work was widely known, Harvey's views, while accepted in principle, wereotherwise overlooked, and their implications forgotten. As a result, the idea of `momentum' withrespect to the functioning of the circulation, has not so far received the attention which it warrants. The present publication sets out to retrieve the situation, and express circulatory activity in termsof the quantity of blood moved, together with the linear velocity of flow, and to demonstrate thedistribution of this energy of motion between the fluid compartments of the body, in a quantitativemanner.

The reason for the comparative neglect of `conservation of momentum' within the fluidcompartments of the body, may not be far to seek. Momentum has two components, mass andvelocity, which are interchangeable in the calculation of relative magnitude, so that a small volumetravelling at high speed, may have the same momentum as a large volume with barely discerniblemovement. Momentum given to stroke volume, may produce a relatively large change in linearvelocity in the small volume of the arterial system, but the same momentum added to the extra-vascular fluid, will produce relatively little percentage change in either volume or linear velocity offlow, and may need a special effort to detect it (see chapter 8). Nevertheless the momentum is stillincreased in the larger volume by addition of a smaller volume with greater velocity, and theincrease may have considerable implication for the function of the cells with which the fluid comesin contact. The interchangeable effects of `volume' and `velocity' in maintaining momentum,become the basis for the interchange between the main fluid compartments of the body, and lies atthe root of circulatory activity, and its relationship with cellular function.

Because the idea of `momentum' is essentially a mathematical one, involving the product of `mass'and `linear velocity', this partition of energy must also involve abstract mathematical concepts. A model of the circulation based on momentum and linear velocity of flow, is for this reason, mostreadily achieved by simple algebraic means.

It would be extremely difficult to assess accurately the total momentum to which a system issubjected in any absolute way. For while the mass of a body can be determined with some degreeof accuracy, the velocity which it has at any particular time cannot. Because of the continuallyvarying influences to which all bodies in the universe are subjected, and the fact that velocity canonly be estimated relative to the motion of other bodies, the effects of gravity, movement of air,temperature variations and convection currents, the rotation of the earth, and so on, make itimpossible to estimate `momentum' in any absolute sense. It is only change of momentum withtime, which can be determined with any degree of accuracy. The idea of `speed' or `velocity' ineveryday use, is based on relating the perceived motion to some fixed point on the earth's surface,to which other speeds can be related. In the living organism, the linear velocity can be calculatedrelative to the position of the body in space, but more conveniently and accurately, velocity in oneregion can be compared directly with that in another, producing a ratio of velocities which thendefines the motion in a particular area, which is directly referable to any motion elsewhere in thesame organism. All that is then required is a convenient yardstick to which these changes in

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velocity (and/or volume, momentum, length, or time) can be referred for comparison. That chosenhere is the ratio of `stroke volume' / `circulatory volume', because it remains relatively stable, andeach parameter can be estimated with a reasonable degree of accuracy, without direct reference tothe circulatory model. It has been named the `circulatory ratio', and it becomes a fixed referencepoint for assessing circulatory function.

The purpose of the ̀ algebraic model', is to help to assess the changes in momentum which occur asa result of ventricular contraction, and the effects any change could have on activity elsewhere inthe body.

Stroke volume then has to be considered in the light of these fundamental ideas. The work done bythe ventricle per beat may be calculated by the product of stroke volume and the pressure at whichthe blood is expelled, including a variable factor representing the kinetic energy of flow of theexpelled blood.

The main part of this work is stored as potential energy in the elastic wall of the aorta in the firstinstance before being transmitted to the walls of the remainder of the arterial tree while it is partlybeing converted to kinetic energy of flow through the peripheral circulation. The relationship ofpotential energy of pressure and the kinetic energy of flow is a recurring consideration, but at thisstage it is sufficient to observe that while the energy transmitted from the ventricle as far as thearterioles is largely in the form of potential energy represented by the arterial pressure, the energyin the remainder of the systemic circulation is for the most part in the form of kinetic energy of flow. Flow energy is provided from the conversion of potential energy represented by the fall in arterialblood pressure between systole and diastole; i.e., the pulse pressure.

The purpose of the circulatory model is to represent as simply as possible, the distribution of energybetween the intra-vascular and extra-vascular compartments, together with the transfer of energyback and forth, and also to show how the energy provided by the circulation is balanced with energyproduced by cell metabolism, to determine at least in some degree, the transfer of fluid across thecell membrane in response to the metabolic activity of the cell. A diagrammatic representation ofthis distribution is set out in Figure 1. In this diagram the central area of energy concentration isthe extra-vascular fluid compartment, from which energy is supplied to aid in controlling the fluidcontent of the tissue cells in response to their metabolism. The energy of the extra-vascularcompartment is derived in the first place from the intra-vascular circulation, transferred to it withthe fluid which leaves the capillaries against the osmotic pressure exerted by the plasma proteins. This energy is kinetic energy of flow, and its amount relative to the total energy of the ventricularcontraction, is controlled by the contraction of the muscular walls of the arterioles, which thereforeregulate the relative amounts of pressure and flow energy presented to the capillaries in the firstinstance, and then transferred to the extra-vascular compartment against the protein osmotic pressureof the plasma. Because of the activity of the arterioles in regulating the relative amounts of flowand pressure energy represented in the peripheral resistance, the regulation of these two forms ofenergy is also transferred to the extra-vascular compartment, thereby determining the energyrequired for balancing fluid exchange across the cell membranes on the one hand, and that requiredfor maintenance of momentum in the extra-vascular fluid on the other. The momentum associated

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with movement of fluid is then returned to the intra-vascular compartment again, where it providesthe kinetic energy of flow for the venous return, the amount of this venous return being related withthe degree of constriction of the venules. A certain energy level must therefore be retained withinthe extra-vascular system at all times to maintain the pressure /velocity relationship, and the venousreturn, as well as the fluid balance with the tissue cells. A large part of the succeeding chapters isconcerned in estimating the relative contributions required from the energy of ventricular contractionfor each of these functions, and their relationships to one another, and to maintaining a stableinternal environment. In an attempt to quantify these relationships, some method has to be foundto represent rather complex variables as though they were simple readily ascertainable quantities. The method adopted to circumvent this problem is to use the simplified theoretical model of thecirculation in which the complex variables are reduced to simple variables for the purpose ofcalculation of energy relationships. The device used is to assume an ̀ average mean' figure for eachvariable.

For example, if we consider the linear velocity of flow in the vascular system, it is clear that this willvary in different areas, from some 200 mm./sec. (or more) at times in the aorta, to 0.5 mm./sec (orless) in the capillaries, and the average mean linear velocity of blood can only be estimatedindirectly. At the same time, the average distance travelled by each ml. of blood in passing fromthe left ventricle to the right atrium will be the mean length of the systemic circulation, representedby `2l'. The average time taken for each ml. of blood to complete the journey will be the averagemean circulation time, and that time will also be equal to ̀ 2l' divided by ̀ v', where ̀ l' is the averagemean length of the circulation and `v' is the average mean linear velocity of the blood. Althoughit would be extremely difficult to estimate any of these variables by direct observation, a reasonableapproximation of their individual values can be obtained through the relationship which they mighthave with other variables having values, which while not known directly, can be related eventuallyto some more readily estimated parameter. For instance, although we cannot measure the meancirculation time directly, it is possible if we know or can estimate the volume of blood in thesystemic circulation with a reasonable degree of accuracy, and the stroke volume and pulse rate arealso known, the mean circulation time can readily be estimated by the ratio of blood volume/ theproduct of stroke volume and pulse rate. The same reasoning applies to `average mean linearvelocity', to ̀ average mean resistance per unit velocity', to ̀ average mean cross sectional area of thevascular bed', and so on with all the other variables in the circulation which we care to represent as`average mean' quantities. Indeed it becomes increasingly clear that estimation of actual values forthese variables is not necessary for them to have usefulness in relating one parameter with anotherin quite abstract terms as one may do in any other mathematical system or relationship.

The use of these symbols however, does not describe the total circulatory system, but remains asimplified model, which may have a qualitative value in describing what could occur in the intactanimal. The validity of the assumptions which this implies can only be tested insofar as theconclusions drawn from the model are mirrored in the function of the intact animal or subject. Thismust continually be borne in mind as we now attempt to explore the relationships thrown up byconsideration of the circulatory model.

No other simple method of demonstrating energy changes and relationships in the circulation seems

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possible at present, and although one must not lose sight of the inherent limitations of the model,the advantages obtained from its use seem to the author to outweigh the disadvantages. But beforeattempting to explore the energy relationships which may emerge between the variables as they aredescribed, it is necessary to list and describe the symbols which will be used to represent thesevariables, and a description of their meaning in each case.

For assistance in understanding the symbols and their interpretations, it might first be best to list thepurpose and meaning of the suffixes which are used from time to time to indicate to whichcirculation or to which part of the cardiac cycle the indicated parameter refers. They are largelyself explanatory.

Suffix Meaning s represents a systemic value p represents a pulmonary value d represents a value at the end of diastole

Now the volume symbols Q represents stroke volume (ml.) V volume of the circulation being considered so Vs volume of systemic circulation (or blood

viscosity) Vp volume of pulmonary circulation Vc volume of the cellular compartment Vx volume of the extra-vascular compartment

Symbols indicating time PR pulse rate in beats per second CT average mean circulation time in the systemic circulation

(seconds)

Symbols of linear dimensions

l average mean length of the blood vessels (circulation). Usuallysystemic value. (Because the venous system is roughly of similarlength having regard to the variability of the capillary circulation andthe relative end of arterial as compared with venous blood, and so therelative length of each system, the venous system is regarded as equalto arterial length, and the total vascular length as `2l'.

L diastolic fibre length of myocardium a average mean cross sectional area of the vascular bed. Usually systemic

value where cross sectional area is proportional to

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v average mean linear velocity of blood. vd mean linear velocity of blood at end of diastole. vx average mean linear velocity of fluid in the extra-vascular

compartment, . or / . LTDTR length-tension development time ratio. The relationship between

initial length of the cardiac muscle fibre and the time taken todevelop unit tension. Provided muscle fibre length remainsrelated to circulatory length, LTDTR is R / PR L/vx

Symbols indicating pressure

AP peak systolic blood pressure. APs systemic peak systolic blood pressure. APp pulmonary peak systolic blood pressure. PP pulse pressure. PPs systemic pulse pressure PPp pulmonary pulse pressure. DP diastolic blood pressure. DPs systemic diastolic pressure. DPp pulmonary diastolic pressure. VP venous pressure VPs systemic venous pressure at end of atrial diastole. VPp pulmonary venous pressure at end of atrial diastole. EDP ventricular end diastolic pressure. ( equivalent to [lactate].L. ) EDPr right ventricular end diastolic pressure EDPl left ventricular end diastolic pressure. R resistance to be overcome per unit average mean linear velocity for each ml. of blood in the systemic circulation, (proportional to ).

`Contractility Factor' (proportional to ) D density of blood. `apparent viscosity' of blood. (proportional to )

The above is an overall list which should allow the identification of symbols which are freelyused in the early parts of the text. Where other symbols are introduced they will be describedand defined at that time.

To conclude this introductory section a list is provided of the relationships between importantindividual variables which have been derived using the algebraic model. The relationships maynot be immediately apparent, but it is hoped they may appear more logical as the account

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continues. The purpose of listing them at this stage, is to give some indication of the scope ofthe ensuing chapters, and of the phenomena which will be examined in each, though notnecessarily in the order set out. At the same time it is desirable to give some regularity to thelist, by adhering to common functional systems with obvious relationships such as the heart, thevascular system, the extra-vascular circulation, the exchanges between cells and their fluidenvironment, and so on, with subdivisions in each group as may seem desirable. As a result,some relationships may be listed more than once where they seem to have significance fordifferent areas, and this may help to coordinate functional activity. Central to the activity of thecells, and therefore to the whole organism, is the cell environment, or extra-vascular fluid, andthe energy which is an integral part of its constitution in the living organism. Commencing withthe relationships present in this region, it is possible to extend them in one direction towards theactivities of the cells, and in the other towards the function of the intra-vascular circulation, theheart, and the function of other organs which are coordinated towards preserving the constitutionand energy content of the cell environment.Extra-vascular fluidVolume, Vx, is equivalent or proportional to each of R.Vs.v/OPP.[ ], D.Vs.v/Dx.vx.[ ],

Vs.v/vx.[ ], v, PR, l.OPP, PPs/PR, and PP/(.l.PR).

Energy contained in Vx, is equivalent or proportional to energy in the systemiccirculation/(OPP.[ ]) , Vs.v.R/(OPP.[ ]), R.vx.v/OPP, Pps/lPR.

Average mean linear velocity of extra-vascular fluid, vx, is proportional to Vs/[ ], and OPP/R. It is also proportional to the ratio

Momentum contained in extra-vascular fluid , Dx.Vx.vx, is proportional to D.Vs.v/[ ],

R.PR.vx, PPs/lPR, vx.R (at diastole), , and venous return.

(Momentum and kinetic energy can only be obtained if volumes are expressed as mass; i.e., asvolume times density. For kinetic energy it becomes D.V/2 of velocity squared. Because allquantities are shown as volumes on both sides of the equations, 'D' and 'D/2' are omitted on mostoccasions where they would cancel out, unless it is necessary to show kinetic energy as a definiterather than a relative value.)

Because of the close relationship which exists between l.PR and Vx, (lPR Vx/l) relationshipsinvolving l.PR, are also important for Vx.

Q/l.PR represents the proportion of ventricular energy which appears as `flow' energy to balanceblood viscosity, or (l.PR Q . Vs)Vs/l.PR is proportional to Q/v, or l.Vs/Vx . APs/l.PR is proportional to R, and APs=R.l.PR.

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PPs/l.PR is proportional to v , and PPs= l.PR.v .

The work of the ventricle per beat available to move blood is Q.R.l.PR, after supplying energysufficient to overcome the peripheral resistance. Volume and energy exchanges between cells, extra-vascular fluid, and intra-vascular circulation.Cell volume, Vc, is proportional to ̀ R', (at systole) to [ ] (at diastole), and is reciprocally relatedto cell oxygen consumption.

Vc/Vx is proportional to R/ , R/v, [ ].0/l.PR.

Vc/Vs is proportional to R/Vs, or . i.e., l. , or cell energy at diastole

Vx / Vs is proportional to / Vs: : / Q or or Vs.v / PR Vs : Vx : Vc :: 0 : v : R . Vx/Vs v/ ; Vc / Vx R/v ; Vc/Vs R/v / R.Vs : Vx / vascular filling : Q / l or or .If Vs is constant, 1 / PR = Vc / Vx .

Energy equivalents for cardiac and intra-vascular function and fluid and energy exchanges withother fluid compartments of the body.

Stroke volume, `Q' is proportional to [ , l Vs, l , ventricular efficiency, ventricular filling,and the venous volume required to produce ventricular filling. It is also directly related to theoutput of the heart, the amount of ventricular work, the linear velocity imparted to arterial blood,the momentum maintained in the circulation, and the energy and fluid exchanges with the othercompartments of the body. Stroke volume is inversely related to pulse rate to regulate cardiac output,to `R' in regulation of ventricular filling, to end-diastolic pressure in the ventricle, and to diastolicand systolic blood pressure. Some of its relationships with other circulatory variables can bederived from consideration of its many energy equivalents, through which it is able to influence allof the other variables which are encountered in the operation of the circulation, from contraction ofthe ventricle, to the volume of fluid exchange which takes place between the cells and theirenvironment. These relationships result because the circulation is based on a relatively constant ratiobetween volumes, velocities, pressures, and the duration of the cardiac cycle with circulation time,which is a fundamental element in its functional design. This ̀ circulatory ratio' limits the size andstructure of the various elements with respect to each other, and allows the exploration ofrelationships between them, by the same algebraic methods which are familiar from their applicationto the physical world. The size of the stroke volume is central to the ratios which constitute thebuilding blocks around which the cell environment is constructed, and which is necessary to ensureadequate nutrition, and fluid and energy exchange, for each individual element. The importanceof the stroke volume lies in its quantitative relationship with the volume of the circulation which itsupplies, to form the volume ratio. This is one of an equivalent series of ratios, all of which havethe same quantitative value, or `circulatory ratio'.

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The equivalent expressions which constitute the circulatory ratio are; Q/Vs = v/lPR = v.R/APs= 1/CT.PR , and these can be expanded to give more complex equivalent ratios such as

so that eventually each variable may be expressed in terms of the others as may be convenient forthe occasion. Each of these expressions is proportional to the mechanical efficiency of theventricle.Returning now to other equivalents involving `Q', Q . APs = work of the ventricle per beat = v . R . Vs v.Vs .D/2 D.PPs / 2Q (of which D/2 can be eliminated as described above, so

that PPs v.Q.Vs.)

Q/Vs venous volume / Vs The volume of the systemic circulation is Vs, or a.l

and this is equivalent to R.Vs is equivalent to a is proportional to 1/ oxygen concentration squared; or Vs / l a =

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Systolic arterial pressure is R . l. PRPulse pressure is v.l.PR or APs.PR.Diastolic pressure is R.lPR v/vd ; or . EDP ; or vd.l ; or v

and the ratio of DPs / PPs = R.vd / (D ) or l / D.v ; i.e., APs / D. or 1/lPRPotential energy is proportional to APsKinetic energy is proportional to PPs PPs / APs v / R Oxygen usage of the left ventricle is proportional to APs.PR.Vs or .Power developed by the ventricle (energy release per second) is proportional to

.

Energy released by the ventricle per beat is proportional to ;Force of ventricular contraction per beat is proportional to l.PR ; and the force developed in unit time to (momentum in tissue fluid at systole). Power transmitted to the circulation is /l, or PPs/l ; and efficiency is proportional to Q/Vs , which is the `notional efficiency' of the circulation.`Power factor' (ventricle) is PR . APs . (equivalent to PPs, or power developed by the ventricle times circulatory length)Mechanical efficiency of ventricular contraction is proportional to EDP .Q / (VPp.Vs) ; or Q . DPs / (Vs . VPp); or vd.R / VPp; or vd / PR; or l .l/L . (The relationship between 'l' representing circulatory length, and `L' or diastolic fibre length of the myocardium, is adjusted by constriction of the arterioles or tone of the arterioles, which varies continuously to maintain 'l L', under the influence of oxygen concentration, hormonal concentrations, and the autonomic nerve supply.)

. Energy which needs to be produced per second is , or PPs.Q ; and external work of the ventricle per second is Q . APs . PR , or power times vascular filling;for a 0 oxygen consumption proportional to R . . initial length of cardiac muscle fibres ; or Vs.APs.PR.VPp/DPs ; where DPs = . EDPl .

The contractility factor, ` ', = vd . lPR / ( v . LTDTR), equivalent to ` ', or

Active State is proportional to . vd / v , or

Inotropic state is proportional to l.PR / (LTDTR . ), or 'l' . Energy transfer factor (ventricle)is proportional to DP/AP or vd/v(as a fraction of LPR) i.e., PR

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Ratio active state / inotropic state (i.e., PR / EDP or ) Pulserate, PR, is proportional to heart volume . EDP . LTDTR / (Q . [lactate] ) ; i.e.; heart volume . LTDTR / Vs; or if l/L remains constant, and PR is then proportional to l . vx..Energy equivalents in terms of gas concentrations and/or concentration of metabolites.Q is [ ;R is [ ] ;a is 1/[ ] ;Vs is ;

is proportional to [lactate]. Vs for normal cell function.

Total power required in system is [lactate] .Vs. .v is ;

; or momentum / APs; or the reciprocal of .

Momentum in circulation is proportional to D.Vs.v ; Q.l.PR ; and to .Momentum added by ventricular contraction is proportional to ; or .

Momentum required for ventricular filling is ; or Q.R; and this is also equivalent to the momentum required for exchange of fluid from the cells, and for transfer to the venous system in order to accomplish venous return.Momentum in extra-vascular fluid required to transfer fluid into the cells is proportional to Q.v/Q.R ; i.e. v/R , but only while v is equal to or greater than . (Because blood may on occasion pass into the capillaries from the arterioles in a pulsatile fashion, this could imply that the linearvelocity of blood in these areas was in excess of `v', the average mean linear velocity of flow, forpart of the cardiac cycle, and this affects the formation of extra-vascular fluid, so that its volumeexpands for part of the cycle at a greater rate than if the linear velocity of flow remained at the`average mean' value throughout. Linear velocity of flow will be greater than `v' in the early partof the cardiac cycle, but less than `v' in the later part of the cycle, but this variation with time islargely eliminated by using the average mean value for `v', so that the final balance with the cellswill be determined by the `average mean linear velocity' , or `v', despite the pulsatile increase involume and linear velocity which occurs. Fluid therefore enters the extra-vascular space, and thecells in a pulsatile fashion, but because `Q' and `R' determine the momentum of fluid leaving thesystem, the outflow is not pulsatile, and enters the veins in a linear flow pattern.)

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APs is proportional to . PPs is proportional to While v / R is proportional to PR ;

v / Q is proportional to ; And Vx is proportional to .

Oxygen transported in unit time is proportional to Q.PR, or ; i.e.; v /

. .

Ventricular Efficiency is proportional to ; or Q,R.DPs.lPR / (Vs.VPp.APs); or Q.R.PR / Vs.VPp (or ventricular filling times energy transfer to ventricular contents, over systemic blood volume times pulmonary venous pressure; which can be resolved into Q.v / Vs.v , equivalent to `l' . (Thesuggestion is that while `v' in the systemic circulation is proportional to diastolic blood pressure,VPp in the pulmonary circulation approaches `v' also, but the actual value depends on `R venous'approaching the value of `R systemic' , and the product with pulmonary linear velocity of flow, orpulse rate to produce `VPp'. This only allows `VPp' to equal `v', if `Rvenous' becomes equal to`Rsystemic', and ̀ L', representing diastolic fibre length in the left ventricle, is directly proportionalto `l', the length of the systemic circulation. The value of `DPs / VPp' is modified as `Q / Vs' isvaried, for example by vasoconstriction.)

FIG 2

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Figure 3.1. Schematic diagram similar to Fig.1., but with energy values selected toillustrate the exchange of fluid volume and energy equivalents between the ventricle,the intra-vascular, extra-vascular and cellular compartments in association with thecardiac cycle.

The stroke volume,`Q', is regarded as directly related with the septal width (commonto both ventricles) which is limited by tension in the fibrous pericardium dependingon carbon dioxide tension maintained in the blood. The stroke volume increasesthe circulating blood volume at each ventricular contraction by the fraction ̀ Q/Vs',the circulatory ratio, but the circulatory volume returns again to its diastolic value,`Vs', before the next cycle. The circulatory ratio is directly related to the gasconcentrations maintained in the tissues, ̀ '. The arterial blood volumeis proportional to ̀ R.Vs/l.PR' (i.e., ),and the venous blood volume to ̀ v.Vs/l.PR'(i.e., l. where both arterial and venous volumes are estimated at diastole). The increased filling of the capillaries achieved by ventricular contraction producesa value for capillary filling proportional to ̀ Q.R.vx', and the increased energy in thisregion available for transfer of fluid and energy to the extra-vascular space isproportional to `Q.OPP'. The added momentum provided to extra-vascular fluidis proportional to `Q.l.PR', (the total momentum provided to the system by thecardiac output, after allowing for the linear velocity increase,`l'). This momentumis absorbed by the extra-vascular fluid, where `Q' is proportional to `Vs'.`l' ,or`vx..l. ', while ̀ v' is proportional to ̀ Vx'. So long as the value of ̀ Q' remainsrelatively constant from beat to beat, `vx' will also remain constant, so that theincreased momentum is absorbed by increased volume of ̀ Vx', as the linear velocityof flow of blood rises and falls during systole, and this constitutes a transientincrease in the ̀ store of momentum' during systole. The ̀ energy balance' across thecell membrane is then disturbed, and energy and fluid passes into the cellproportional to ̀ v', but dictated by the passive permeability of the membrane, whichis proportional to ̀ ', to become equivalent to ̀ Q.R' The momentum transferredacross the cell membrane is then proportional to ̀ v. ', which must be equivalentto `Q.R', if stroke volume is to be maintained proportional to the amount of fluidtransferred. This means that `v' must be numerically proportional to `R squared',and the cellular volume increase (which must be proportional to ̀ R') requires ̀ v' toincrease as ̀ R squared' to increase the energy ̀ store' in the cell, proportional to thevolume increase. (Equivalence between the values of `R' and `Q' is only possible ifthe oxygen concentration remains constant, so that `Q/R' and `v/ ' are alsoconstant. These restraints require ` ' to remain relatively constant in order tomaintain adequate cell and circulatory function).

The ̀ active permeability' of the cell wall depends on cell oxidative metabolism, andis a function of `[ ].l', which is directly proportional to `Q'. The extra

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momentum which has been added to the cell, and is proportional to `R', togetherwith the stroke volume, `Q', returns fluid to the extra-vascular space, (where thefluid volume has diminished because the linear velocity of the blood has fallen froman average mean value ̀ v', to the diastolic value ̀ vd' at the end of the cardiac cycle).Because of this fall in linear velocity, fluid returns from the extra-vascularcompartment to the intra-vascular compartment, and the amount of this fluid will bedetermined by `Q.R' which is proportional to`R.vx' times [lactate] or the osmoticpressure exerted by the plasma proteins and residual energy in extra-vascular fluid(while `Q/Vs' remains constant)The amount of momentum equivalent to `Q.R' iseventually transmitted to the ventricle where it is absorbed to produce `ventricularfilling' for the next ventricular contraction.

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Figure 3.2. Schematic diagram to illustrate the relationships of linear velocity ofblood flow, the force of ventricular contraction, and the `free energy' levelsmaintained in the cells, with the overall level of `lactate' present in the tissue fluid.

The linear velocity given to the blood by ventricular contraction depends on theforce of the contraction which is proportional to `l.PR', and this is a function of thelength of the interventricular septum, (and the diastolic length of myocardial cells)common to each ventricle. The linear velocity of flow determines the pulsepressure (`v.l.PR.D/2'), and the momentum of the circulating blood (proportional to`v.Vs). A proportional amount of this momentum is transmitted to the extra-vascular fluid to increase `Vx', and `vx', and it requires an increase in the `free'energy level within the cell which remains proportional to `l.PR', to maintain thefluid and energy exchange (see fig.2) providing ̀ vx' remains unchanged. The pulserate must then be matched by an increased rate of energy release in the cell from theenergy `store', or from glycolysis, if the fluid exchange, and so the production ofcarbon dioxide, is unaltered. In other words pulse rate is equivalent to the rate ofenergy release from high energy phosphate bonds (or from glycolysis) whichconstitute the ̀ transient energy store' in the cell. At the same time the energy storeneeds to be maintained by regeneration from `lactate' in the cell fluid, whichincreases fluid exchange, `free energy' and `energy store' equally when the level oflactate is raised. The fluid exchange, `free energy' and `energy store' are allaffected equally by the lactate concentration, their individual values depending onthe individual values of `[ ]', `[ ]', and `PR'. The individual values of `Vc',`Vx', and ̀ Vs', are related by values of velocity and momentum, viz., ̀ R', ̀ v', and ̀ Vs',while ̀ Vx/Vc is proportional to ̀ v/R'; ̀ Vx/Vs' is proportional to ̀ v/Vs', or ̀ v /vx';and `Vc/Vs' is proportional to `R/Vs', i.e., `l. '. As the extra-vascular fluidvolume contracts when `v' falls to `vd' at diastole, and fluid re-enters the intra-vascular compartment, `vd' has an important part in maintaining venous pressureand venous return so that ̀ Q.R' is maintained, and the levels of ̀ R.vd' , and ̀ VP' arealso affected, while ̀ R.vd/VPp' must be maintained to maintain ventricular efficiency(chap.6).

Figure 1. is an attempt to set out the main features and conclusions of the circulatory model as an overallguide to the concepts which are developed in the later chapters.

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Figure 3.3. Schematic diagram to illustrate the equivalents of ventricular filling, theenergy transfer from ventricular wall contraction to the ventricular contents, the internalradius of the lateral ventricular wall, together with the energy stores in the aorta, theextra-vascular fluid, and the individual cells, the internal resistance offered by the bloodto linear flow, and the momentum equivalents of the `energy storage' in eachcompartment.

Ventricular filling `Q.R' is the momentum equivalent which blood returning to theventricle must have, in order to allow it to distend the ventricular cavity, against theresistance offered by pericardial tension. Of this momentum, ̀ Q' is related to the widthof the common interventricular septum, while `R' is related to `energy transfer fromventricular muscle contraction to ventricular contents' , by the relationship it has with`end diastolic pressure', and diastolic blood pressure, which is proportional to `EDP'times `contractility' (chapter 6). The `energy transfer' depends in turn on the internalradius of curvature of the lateral ventricular wall (chapter 8). Increase in themomentum equivalent which has to be overcome by each ml. of blood per unit velocity,or `R', increases storage of energy in the aorta with increase in systolic arterialpressure, `R.l.PR'. The vascular filling `Vs.R', is also increased, together with theamount of energy which has to be maintained for an adequate circulation, `Vs.R.l.PR',while the momentum in the extra-vascular circulation must also be increased to keep ̀ v'and `Vx' numerically equivalent to `R squared', and to maintain an adequate level offluid and energy exchange. The peripheral resistance is `v.R', and this means thatmomentum in the extra-vascular compartment must increase as `R cubed' , while themomentum passing into the cells increases as `R squared', of which `R' is contributedto increase the energy `store', while the `free' energy (`l.PR') and `work capacity' (`lsquared . PR') in the cells must also increase proportionally as `R squared' in order tokeep the fluid and energy exchange at its previous level, i.e., equivalent to `Q.R.vx' , or`Q.OPP'. Although the momentum required to overcome resistance in the intra-vascular circulation has only increased as `R', the equivalent increase in the energycontent of the cells has increased as `R squared' , and in the extra-vascular fluid as `Rcubed', with an equivalent increase in cardiac work, to maintain the level of `storedenergy' in each of these regions. The energy equivalent of `R' is `l.[ ', and thismeans that `R' can be increased by increasing `[ ]' in the cells, and so the externalresistance offered by constriction of arterioles; or by altering the internal resistance toflow in the blood vessels, (mainly the veins) when the apparent viscosity, ` ', isincreased by altering the oxygen saturation of blood. At the same time the blood vessels(veins) may have their diameters increased by relaxation of muscular walls, which thenoffer decreased external resistance to flow, though the overall resistance may beincreased through increased viscosity. The increase in viscosity only affects flow in theintra-vascular compartment, because in the extra-vascular compartment, the viscosity(while not negligible) approaches more closely that of water. The viscosity of bloodassumes considerable significance for regulation of glycolysis, and of the energy

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balance in the effector cell.

Control of glycolysis depends on the ratio between the oxygen partial pressure in thecells and that in the blood, as well as the level of lactate persisting in the cells. Thebalance between oxidation and glycolysis is regulated by the competition for inorganicphosphate, essential for continuing glycolysis, which is inhibited when inorganicphosphate is not available in sufficient concentration.

The oxidation of pyruvate (lactate) can lead in either of two directions. a. Oxidative phosphorylation' or the production of energy which is then transferred intohigh energy phosphate bonds, and increases the energy `store', also competes forinorganic phosphate, and inhibits further glycolysis. b. Oxidation via the Kreb's cycle to produce carbon dioxide and water , but with theresulting energy directed to giving kinetic energy to fluid leaving the cell in the `fluidand energy exchange', and eventually transmitted to the venous return. It can continuein the absence of inorganic phosphate, but with reduced glycolysis. With a greateroxidation rate, the oxygen content of the cell is reduced, decreasing the ratio of

cells/ blood. When cell oxygen falls, blood oxygen concentration can only bemaintained by increasing linear velocity of flow with incomplete equilibrium betweenblood and cells. If the linear velocity of blood flow is diminished, the differencebetween the two areas becomes less pronounced, and cell oxygen is maintained, while of blood falls, with an increase in the internal resistance to flow, or `l. ', as viscosityincreases. Continuing glycolysis becomes related to blood viscosity, which is increasedbecause of increased oxygen usage in the cells, but glycolysis must increase to maintainvelocity of blood flow. Glycolysis and lactate production only continue, while incells remains at a sufficiently high level and carbon dioxide concentration is reduced toreduce viscosity of blood. As viscosity increases, more energy is needed to maintainthe energy exchange, and stroke volume. Oxidative energy to maintain carbon dioxideis reduced as cell oxygen concentration is increased, and is supplemented by glycolysis,which increases when the oxidative enzymes are inhibited with the raised oxygen levels.Partition of oxidative energy between `energy exchange', and `storage', is partlyreplaced by glycolytic activity, and as a result, glycolysis is increased, and viscosity isovercome.

For `pacemaker' activity e.g., in nodal cells. As `v' increases with respect to `R', thecell volume increases, and the cell elongates at a greater rate than the dielectricconstant of the cell membrane, which is related to oxygen partial pressure in the cell. The capacity for spontaneous depolarisation is increased, together with pulse rate . If on the other hand, `R' increases with respect to `v', the dielectric constant of themembrane increases with respect to cell volume, and the pulse rate may decrease. If`v' is reduced too drastically with respect to ̀ R', spontaneous discharge of the polarisedcell may cease altogether.

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The increase in the value of `R' has far reaching significance. To maintain the energyand fluid balance within the cells, momentum needs to increase as the ratio `v/R', andthe increase needs to be sufficient to maintain a balance between `v' and `R2' so that `v'is proportional to `R2'. If `v' increases at a more rapid rate, there is relative increasein cell volume, and increased length of muscle fibres, but with a relative reduction of theenergy ̀ store' in the cells, (i.e., relatively less A.T.P.) and a relative reduction in the rateof cell membrane polarisation with respect to cell volume. The `free energy' level hasincreased in the cell more than the dielectric constant, and spontaneous depolarisationis assisted. Should `R2' increase relative to `v', the dielectric constant of the cellmembrane increases with respect to cell volume, and depolarisation occurs less readily.As `v' increases , `free energy' (i.e., `l.PR') is also increased, and this assistsspontaneous depolarisation, while as `R' (and ) increases with respect to `v',spontaneous depolarisation is delayed. In this way, pulse rate set by the pacemakercells may be either increased or decreased, depending on energy levels which persist inthose cells.

The volume of extra-vascular fluid,`Vx', is proportional to `v', and to `R2', and `filling'of the extra-vascular space, `Vx.R', is proportional to `R3', which affects the energyexchange with the cells as follows:- Of the energy supplied, a portion,`R.Q', is themomentum needed for a volume `Q' to enter the cell by overcoming restrictedpermeability, (proportional to ̀ R', or ), which allows the energy store in the cell toincrease as `R2'. The stored energy in the cell is proportional to `R', leaving anotherfactor proportional to `R', to be given to the fluid leaving the cell as `v' diminishes withdiastole, and the fluid leaving the cell needs the further momentum equivalent, ̀ Q.R', toeventually overcome the resistance to ventricular filling. The volume of this fluid, ̀ Q',needs a further velocity factor equivalent to the linear velocity of the extra-vascularfluid, or ̀ vx', so the energy needed and made available by cell activity, becomes ̀ Q.R.vx',or ̀ Q.OPP', which is the energy needed for fluid to enter the circulation (venous system)once more. Now `vx' is proportional to `Vs', and the extra-vascular fluid volume atdiastole is ̀ Vx(d)', which is proportional to ̀ vd'. The intra-vascular momentum is then`Vs.vd', proportional to venous momentum (`V venous . v venous') and `v venous' isproportional to `Vs . vd'/ `V venous'.

Because `V venous' must be less than `Vs', `v venous tends to increase with respect to`vd', and venous blood is accelerated with respect to general intra-vascular velocity atdiastole, and this allows the increased velocity needed for the venous return, whichcomes from the effector cells (and originally from ̀ v') by way of transient energy storagein the cells.

The increase in the ratio `v/R' would usually require reduced glycolysis, because therelative fall in the value of `R', reflects a relative drop in A.T.P. and an increase ininorganic phosphate within the cells. As glycolysis increases the concentration oflactate, there follows an increase in oxidation, and in the production of carbon dioxide

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(as the inhibition of the oxidative enzymes is reduced by the added lactate). Theincreased energy produced appears as extra A.T.P., with altered cell permeability, anda relative change in energy storage (and of `R', depending on oxygen concentration). These reactions all balance each other, and maintain steady values of all theparameters for a given activity level of `effector cells', and ventricular muscle cells, byregulating the linear velocity of fluid flow in each area.

Within the active `effector cells', the peak value of the `transient energy store' withsystole, is proportional to ̀ R2', and also to ̀ Q', representing the fluid which has enteredthe cell from the extra-vascular fluid. The capacity of the cell to perform `work' isproportional to `force of contraction' times `circulatory length', or `l2.PR', afterovercoming the internal resistance to flow (`l. ') for a given value of ̀ Q/Vs'. The ratioof `transient energy store'/`work capacity' is equivalent to [lactate].R.PR/`freeenergy'.`l'.l.0' , or `l2.0.R'/`l3.0.PR'. The ratio `R/l.PR' represents `peak energystore'/`work capacity' , or energy balance in the cells, and also the energy balance in theventricle between the `momentum equivalent' representing `ventricular filling', `Q.R',and the `momentum of the venous return', or `Q.PR'. The energy balance between thetwo regions only holds while [lactate] is proportional to `R /Vs 'in the systemiccirculation. For a constant pulse rate, the energy developed by the ventricle per beatis proportional to [lactate]. , or . Of this energy, ̀ l' represents the energyrequired to raise the linear velocity of venous blood, to the `average mean' linearvelocity of blood in the appropriate arterial circulation, while ` ' is

proportional to , the energy contributed by passive permeability, and cell volume,and representing the ratio of ̀ stored energy at systole` with maintenance of cell strengthand blood volume.

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Summary.

Ventricular contraction gives momentum to the blood, and the energy providedhas two functions, to overcome resistance to flow, and to ensure that sufficientmomentum is retained to provide ventricular filling when blood is returned to theheart. The idea of momentum has not been greatly stressed previously, but itsimportance lies in allowing larger volumes with low velocity to have the sameamount of motion, and so kinetic energy, as a smaller volume at greater velocityof flow. The movement of the larger volume and the energy it contains tends to beoverlooked. The relation is a mathematical one, and is best expressed inNewtonian concepts of space and time, and in algebraic form, to quantify potentialand kinetic energy in the main fluid compartments and the exchanges betweenthem. Momentum finally is returned to the venous system, with importantmodifications resulting from cell activity. Complex variables are reduced to simpleform by expressing them as ̀ average mean' values. A theoretical example is givento illustrate the value of the simplified model.

A list of symbols which represent different parameters used in the algebraicexpressions is included, along with a number of derived expressions whichconstitute the model, though their derivation and application are not considereduntil later sections.