Simulation of coronary circulation with special regard to the venous bed and coronary sinus occlusion

Simulation of coronary circulation with special regard to the venous bed and coronary sinus occlusion W. Schreiner,

F. Neumann

and W. Mohl

Second Surgical Department, University of Vienna, Vienna, Austria

Received May 1989, accepted October 1989

Abstract 7?u vasm?ar beds of the left circumjex and the lej anterior &cending cos0n.q

arteries are modelled by means of coupled differential equations that considcr an arterial, a capillary and a venous section. In a stepwise procedure, experimental o!atajom normal corona7perfkion and c0r0n.u7 sinus occlusion are used to assess the modelparameters. For venous distensibility, a non-linear form ofpressure-volume relationshipprovedvital to reproduce the characteristics of the rise in venouspressure afler the onset of corona7 sinus occlusion. Numerkal integration was carried outfbr normal perfusion and fir corona7 sinus occlusion, yie&iing time courses offlws, volumes and pressures within large corona7 arteries, capillaries and corona7 veins. Corona7 sinus occlusion reduces total mean flow by 18% and divio?es intramyocardial flow between the capillartes and the veins into a fixward component of 3.03 ml/ and a backward component of - 1.54 ml&. l’iiis result represents a prediction fir a haemoa’ynamic quantity which is therapeutically important but inaccessible to measurement. Varying a2gree.s of systolic myocardial squeezing are studied to dtkphzy the impact of myocardial contractility and vessel collapse on the mean values and phasic components of intra-myocardial flows. Keywords:

Numerical simulation, mathematical modelling, coronary circulation, haemodynamic parameters

INTRODUCTION The redistribution of blood from the epicardial coronary veins toward the regional myocardial capillary bed during ressure-controlled intermittent coronary sinus occPusion (PICSO) and during synchronized retroperfusion (SRF’) is assumed to be of great thera eutic importance. Therefore, much experimental Bata have been acquired under these interventions l-7 and the consequences of more eneral work on coronary haemod amics8>g has coronary sinus %een evaluated for the optimization oiyn interventions “7 “. Additionally, phenomenological mathematical models have been established and tested to control PESO automatically12-15. As a supplement to these studies, the present simulation provides a mechanistic explanation of coronary sinus pressure dynamics and provides an insight into intramyocardial haemodynamics. This mechanistic a preach was established to satisfy the following dp emands: 1. To reproduce the pulsatile venous pressure dynamics found experimentally under coronary sinus occlusion, which has not been covered by previous mechanistic models; Correspondence and reprint requests to: Dr Wolfpg Schreiner, Second Surgical Department, University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria 0 1990 But&worth-Heinemann 0141-5425/90/050429-15

2. To estimate the basic redistribution of intramyocardial flow g etween capillaries and veins under coronary sinus occlusion, which is inaccessible to measurement; 3. To reproduce well-known features of the arterial part of the coronary circulation, a topic that has also been covered by other modelling approaches. THE MODEL Description of compartments The ostium of the left common coronary artery (LCA) is assumed to be perfused at a pressure P=*(t) which constitutes a driving force within the model and which is obtained numerically by Fourier expansion of the measured aortic pressure (80 < PLcA< 125mmHg) I6 . Traversing the small resistance R, (with a corresponding pressure drop RwAQLCA), the total left coronary blood flow bA then divides (at a bifurcation site with pressure P& into the left circumflex (LCX coronary artery bed (resistance I REX %+c,) and the left anterior descending art-mp,flow artery (LAD) & ed (R2%+,, QkLJ. In the LAD branch, an additional resistance R!&? can be inserted to simulate LAD-stenoses (RZ?>O). Collateral flow is not considered in the model. The capillary compartment is modelled separate1 but in the same manner for both the LAD and LC x

for BES J. Biomed. Eng. 1990, Vol. 12, September

429

Simukztimr of wmnary dmlation: W?S&w

et al.

cop )ven

LAD

I Coronory arteries

p

LAD

t

sqz

Venous comportment

Copillory comportment

Right otrium

Figure 1 Schematic diagram of the model. The following components (from left to right) constitute the vascular bed of the left coronary arte in the model. From the let? coronary artery ostium (driving pressure I+.&)), the left common artery (RL~A, Qm$, leads to the bifurcation (PM7 . LCX and LAD branches are modelled similarly. The LCX coronary artery (R%&,, Q%$+,) leads to its capi%compartment (Cz, Vz, Pgx, which is subjected to extravascular squeezing pressure (Pa. The venules draining the LCX bed (Rz,, Q,“_-) and the LAD bed (Rz,, Q&!!.-) lead to one single coronary venous compartment (volume-dependent compliance &,,( V&), volume V_, Yen). The venous compartment drains to the right atrium (PM = const) via the coronary sinus (Rven+~, Q--M) and the extra outlet (R,, Q& shown as the narrow outlet above the coronary sinus

lines. Both compliances (C@’ and (22) are made to be equal and constant. Volumes (Vg$x, V$$‘) and pressures (PZ:, I%&!‘),being calculated quantities, are on1 equal in the symmetric case, i.e. in the absence of IY24 D-stenosis. Both capillary compartments are subjected to phasic extravascular intra-myocardial squeezing ressures (also known as intra-myocardial pressure) (5 &)), re presenting a second driving force in the model. The key role of squeezing in intramyocardial haemodynamics was introduced in early theoretical investi tions’ and was found to yield realistic coronary 8”ow patterns in model calculations, if one assumes proportionali to the left ventricular pressure, PLvg. Experimen d work confirmed that phasic myocardial squeezing does indeed run parallel to PLv17-rsand is very close to PLYin subendocardial layers. Therefore, using a Fourier expansion of PLv, squeezing pressure can be defined as: LCX-bed:

Pr

= ynO,,,, PJt)

(la)

LAD-bed:

Py

= ysten Y,,~~PLv(t)

(18

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J. Biomed. Eng. 1990, Vol. 12, September

Experimental datalg indicate that 75% of PLv corresponds to a s ueezing pressure experienced around a fractional w3 1 thickness of 0.7-0.8. Hence, for a normally contracting myocardium with e ual squeezing in the LAD and LCX area, the vaues y,, = 4 0.75 and ysh = 1 are selected (setting ye,< 1 would simulate a reduction of function in the LAD-dependent region of the myocardium). The outlets from both capillary compartments, described b resistances (R&,,, R&$2+,,) and flows drain into a single venous com( Qi&en, d &!LJ, partment common to both the LCX and LAD beds. Venous pressure Pvenand venous volume V.,, are linked b a non-linear relationship (see next section for detaiL of venous compliance). The venous compartment drains into the ri t atrium via two pathways: (i) an ‘extra outlet’ (Bh ow resistance R,), which represents the flow in smZf1:: shunting veins draining directly into the atria; and (ii) the main outflow path via the coronary sinus (flow Q_._lu\, resistance R _._&. Coronary sinus occlusion is

Sinrcllationofconmaty dn-ukation: W: Sduiwr et al.

simulated b setting R_-rRA = a, corresponding to zero flow. l-Ke right atrium is assumed to be a reservoir at constant pressure, PRA.

Relationshipsbetween parametersand calculated quantities Rather than presenting the model equations in closed form, the relationships will be discussed first and then all the components (inflow, outflow and volume changes) assembled in the final equations given in the next section. Assuming the volumes and flows in all compartments have certain values (initial conditions), corresponding pressures can be calculated as shown below. The bifurcation pressure is given by:

Pbif= p,,(t)

- &c,@,Z&

+ QZ&)

(2)

where the term in parentheses is the total left coronary artery flow, QLCA.Intravascular capillary pressure, PKx, is the sum of the elastic reaction of the vessels against distension ( w/ pTx) and the superimposed myocardial squeezing Pgzx: Pgx = E$x/C!$x P$q=

+ Pgx, and

vyc~+Py

Q&EC, = (Pbif- P&‘)/(REcap +p[ Y!&x]-z)

(3b)

(4

corresponding to a (volume-dependent) compliance, defined as the derivative of V,, with respect to Pv,,:

which reduces retrograde flow to zero as the capillary volume approaches zero. The constant /3 = 1 (mmHg s-l ml-‘) re-establishes correct dimensions. The inverse quadratic dependence on V%x has been chosen according to Poiseuilles’ lawi6. Thus the impact of myocardial squeezing on coronary haemodynamics is modelled directly on the decrease of capillary volume rather than introducing the concept of back-pressures as the driving forces. In fact, the inverse quadratic term in equation (6b) represents a pulsatile ‘extravascular resistance”’ superimposed on a static component RSZcap. The throttling of flow following vessel collapse is often referred to as the ‘waterfall phenomenon’21 and emerges naturally from the present model. For the LAD bed, e uations and parameter values are similar (R&&, = % !&3,,), except that RZ,, + R#? applies, instead of RkECpp alone. In the current study, LAD stenosis will not be considered, i.e. RW=O.

Flowj-on the capillaries to the venous compartment. The flow from the capillaries to the veins is again governed by two different equations. For a positive pressure gradient (Pgx - P,,,, > 0) a ‘forward form’ equation is used, which reduces (forward) flow as the capillary volume approaches zero.

Q

where xv_, is the reference compliance (mlmmHg-‘), u is the slope of change in compliance (ml-‘) and Vte, is the reference venous volume (ml). Equation (4) is chosen in such a way that, as the venous volume increases, the compliance diminishes, i.e. there is an increasing elastic reaction against further venous distension. Subsequent sections and the Appendix will discuss this issue further. If all the pressures are known, then the flows, i.e. time derivatives of volumes, can be calculated as follows.

Flowfrom the corona arteries to the capih!ariRF.Q$& is proportional to z e gradient between bifurcation ressure and capillary pressure and may be either ! or-wards (i.e. $%,,>O) or backwards. A positive gradient Pbir- & &X leads to the ‘forward form’ of equation (6): (64

A negative gradient induces retrograde flow. However, backward flow ceases as the capillary volume

LCX capeven

=

(Pk,” - Pv,)/(R@+,n +P [ V!$-“)

(7~)

If the gradient is negative, the ‘backward version’:

Q

Q%Lp = (Pbif- Pk$x)/RSZcap

(64

(3a)

Venous pressure at any time t is calculated from the pressure-volume relation: P Ye”= v,,, xi,‘. e”(Len-?m)

sure gradients, equation (6a) has to be substituted by a ‘backward version’:

LCX cap-ven

=

(Pk$’

-

Pv,,)/(Rk$+;,,,+ p [ Ken]-2)

(W

applies, curbing retrograde flow if the venous comartment is emptied. The same equations apply to the !A D bed (superscripts LCX-, LAD). Flow from th venous bed to the right atrium. The right atrium is considered to be a reservoir of infinite volume at constant pressure Pm. Hence, only the forward flow must be restrained if the venous volume approaches zero (ahhou this is very unlikely, this case is included for p e sake of completeness). Therefore, for a positive ressure gradient Pv,,, - PM ~0, we have the forwardpversion:

= (L - Piu)~(&m+wq?jtLn]-‘) Q ven+~~

(84

whereas for a negative gradient the backward version applies:

= (Ken- PRA)DL+RA Q ven-RA

tW

Extraordinary Jaw out of th venous system. Qex represents non-coronary sinus drainage from the venous compartment to the atria. Since these pathways originate in smaller veins within the myocardium and drain to both atria, there is a spectrum of pressure gradients (with its lower end equal to P,, - PM) which determines Qex. As an ap roximation, a constant pressure offset P$$ is add d; to P,,,. Moreover, QExis set equal to zero, except for positive pressure

J. Biomed. Eng. 1990, Vol. 12, September

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Simulation

ofammycirculation: W.Schcincr et al.

gradients applies:

P,,, - PM, where the forward version

Qex= [I’ve.- &A + Pt)/(R,+/s[

Vi,,]-2)

(9)

Obtaining the time courses of haemodynamic quantities Conservation of mass (i.e. blood volume) demands a balance between changes in volume, inflow and outflow for each compartment: Capillaries LCX: $

VS; = QkE,

Capillaries LAD: $

R$? = QkiE, - Qk$L

Venous volume: $

Fr,, = Qk&Ln + Q$!Ln -

- Qi$iLn (104

- Qex Q ven+FtA

(lob)

(11)

Numerical integration of these differential equations (Euler’s method) yields the time courses of all the haemodynamic quantities of the model. Since the results proved indistinguishable for time ste s At = 0.001 s and 0.0001 s, At = 0.001 s was used 51roughout this work. As an initial condition, all calculated quantities were set equal to one in their respective units (see Table 3), except for PUS= 1OOmmHg and the driving forces P&t) and P+(t) given by Fourier ex ansions. After several heart beats, a periodic soPution was obtained. The heart rate (which was introduced into the model via the driving forces) was set to 60 bpm in all simulations. Usin this heart rate, flows in ml& are numerically equaY to single beat volume increments in ml. TWO PASS-MULTI-STEP MODEL PARAMETFW Direct a ptioti mental values

parameter

ASSESSMENT

OF

settings from experi-

The resistance of the left common artery was calculated from length and diameter16 as RX* = 0.05 mmHg s ml-‘. The resistance ratio Rk$L,,,,/ REC, was set equal to 0.1 (ref. 22), which places the major origin of resistance within the arterioles. The myocardial squeezing was set at 75% left ventricular pressure (Y,,~,.,,, = 0.75) 17-lg. The venous residual volume Een was set at 25m123. The two driving and forces, aortic perfusion pressure (P&t)) myocardial squeezing (x”,,.,,,Plv(t)) entered the model as Fourier expansions of experimental values’6, scaled to a heart rate of 60 bpm. Right atrial pressure (PM) was set at 5mmHg. Preliminary parameter settings It is important to note that the static resistance RLCX cannot be finally assessed from ca,,+ven + RitEcap the pressure gradient (Pm* - PM)/ f.I&Abecause capillary squeezing additionally impedes (&CA) (see Figure 6). Hence, e erimental values for overall resistance either inclu“xe the squeezing component24

432

J. Biomed. Eng. 1990, Vol. 12, September

or are associated with long diastoles in order to eliminate the systolic squeezing effect22. In the light of these difficulties, we adopted a preliminary setting for RLCX cap-rven + RkExp corresponding to a reasonable pressure gradient (100-5 mmHg) at a flow of 200 mlmir+ (= 3.33 ml s-l). For e ual LAD and LCX lines in parallel we obtained (1 $$i+ven+ Rk%,J = 2 X 95i3.33 For a resistance ratio R~L,,/ = 57mmHgsmP. R&p = 0.1 (set a priori) this yields R@L,, = 5.2zHgsml-l and R!$Z,,, = 52mmHgsml’. In a next ste , the following parameters (which refer to the capi lYary and venous sections) were set to ‘preliminary values’, since measurements re ox-ted in the literature are either missing or contr adp ictory. Venous outflow resistance (&,,,+a~) was set at 2 mmHgs ml-l and the extra outlet resistance set at R,, = 50 mmHg s ml-l. Capillary compliance (C!$P = GPx) was set at 0.05 mlmmHg-‘, which is a compromise between the diverging values found in the literature25’26. Venous compliance was made constant (a = Oml-l) at five times the capillary value23: xven= 0.25 ml mmHg-‘. Finally, the venous pressure offset (Pte”n) was set at 5mmHg. Resistances determined from normal perfusion conditions During normal perfusion (i.e. with the coronary sinus open), the mean flow ( A) in the left coron arte is approximately 2P Omlmhrl (or 3.33 ml s-l “r , whitx is 85% of a total representative human coronary artery flow of 250 mls-’ (i.e. 5% of a 50OOmls-’ cardiac output)27. The model was run repetitively: while using the settings described above for the remaining parameters, the sum of static resistances (RYZ,,, + RkE,.,) was changed under constant ratio s R@LJR!&?& = 0.1) until a total flow of 200 mlmin’ was obtained in the resence of squeezing. This resulted in RK,, = 1 33, = 27 mmH s ml-’ and Ii&!!+,, = R@&,, = 2.7 mmHg sml-I, wa ich is well below the preliminary setting based on pressure gradient. Due to symme , e ual resistances ap ly to the LCX and LAD beTlJs. ote that all the otlYer preliminary settings (Rve,,+m, REX, capillary and venous compliance) had very little impact on the total coronary flow during normal perfusion. Extra outlets resistance determined from conditions of permanent coronary sinus occlusion Experimental measurements 12-14* 28 show that mean arterial flow declines by lo-30% near the end of a long (20s) period of coronary sinus occlusion. This fact was used to assess the model parameter REX in the following way. Rven_~ was set to infinity lo6 mmHgs ml-l), In this state, the extra outlet resistance REX) takes over all the outflow normally passing through the coronary sinus. REX was then adjusted until the arterial flow (calculated within the model) was 18% below normal values, yielding REX= 20 mmHgs ml-l (Tubh 7 and 3). This assessment of REX was additionally checked against the fact that non-coronary sinus flow during normal perfusion has been measured to range between 1Oo/o27 and 25Y0~~of total flow. The model yields 10% (Table 3).

Simulationofcmwmy circukation:W. Schei~ et al.

Parametersdetermined from intermittent coronary sinus occlusion conditions The resistance of the venous outlet (Rven_.& was adjusted to yield the rapid and total decline of P venlsrJ within one or two heart beats after the coronary sinus occlusion was released, as observed in animal experiments 28. The result was Rven-+~= 0.6mmH sml-l (Table 7). In the 1 nal step, our experimental data on the rise of coronary sinus pressure 2 l4 were used to assess the capillary corn liances Ce = CkGx and the venous compliance, Befined by xv=,, and a”,. (These had only been set on a preliminary basis during the preceding steps.) These parameters have little impact on total flow in a steady state, and hence a preliminary setting proved tolerable during the assessments described above (see Final pass of adjustment section below). Furthermore, it became evident that C&!’ = Cgx, ,+n and u W” could only be assessed simultaneously, since each of them drastically influences the pattern of pressure. The followin experimental adjustment: 1. Venous pressure should reach plateau values within several seconds after the onset of coronary sinus occlusion; 2. The plateaus should be between 80 and 100mmHg for systoles and around 20mmHg for diastoles; 3. Systolic and diastolic envelopes of coronary venous pressure typically rise with an increasing p, as described earlier in our phenomenof+ ogical models. These three simultaneous conditions impose rather severe restrictions on the three parameters C&$x,xven and a, since the typical form of rise in venous pressure is ve sensitive to any parameter change. Hence a set o parameters which yields the correct pressure rise can be considered as being well defined. This issue is given further interpretation in a later section.

7

Fiial pass of adjustment Starting with the preliminary parameter settings discussed earlier, the whole assessment was repeated, using the results from the first pass instead of reliminary settings for the respective parameters. !k ese ‘second-order corrections’ were small, and the final parameter estimates are listed in Table 7.

SIMULATION RESULTS Time course of calculated haemodynamic

quantities

Starting from initial conditions at t = 0, numerical inte tion is carried out under normal perfusion con Yitions to achieve a eriodic solution after approximately 10s (i.e. 10 K eart beats for a heart rate = 60, see Tabk 7). At t = 20 s, coronary sinus flow is reduced to zero (coronary sinus occlusion) for the subsequent 10s. At t = 30 s, the coronary sinus is

Table 1 Default set of model parameters. All simulation results in this work refer to these parameters unless otherwise indicated (parameter studies). (‘Source’ indicates the method of assessment of each parameter) Parameter

Unit

Value

R LCA

mmHg s ml-’

REX art-cap= RbE,

mmHgsml_’

@.cx cap-“en= K&en

mmHg s ml-’

Source

0.05

Estimated27

27.0 2.7

Sum adjusted for 1

Q LcA

R cap-uen~Rart-p

None

0.1

See references 22 and 27

R “en-P._4

mmHg s ml-’

0.6

Adjusted to yield correct decline after release of CSOL3.14

R,X

mmHgsml_’

20.0

Adjusted for Qwc during CSRz3 and decline of mean

mmHg

5.0

Adjusted as for R,

None

0.75

See references 19 and 44

ml mmHg_’

0.2

Adjusted for Pvenlsys-

p

ml mmHg-’

XW”

1.0

12-14.28 “en1&a

Adjusted for (K&0

Pve+ and pw. rise envelopes”-‘4~28 ml-’

u”e” FL HR

0.3

Adjusted as for

ml

25.0

XW. Preset

bpm

60

Preset

reopened, and the system rapidly returns to normal perfusion. All calculated quantities are plotted in time steps of 0.01 s and the numerical integration runs on 0.001 s increments (only every tenth result is lotted). In the following, normal perfusion will be ax dressed as ‘coronary sinus release’ (CSR), as opposed to coronary sinus occlusion (CSO). In Figures 2-4 and 6, the main charts (a) dis lay the whole time course, while the subcharts (bP and (c) show single beat resolutions during CSO and CSR, respectively. The phasic location of systole and diastole is shown in Figure 3c, which virtual1 reflects the driving force P,“(t) (see equations P1) and (3)). Due to the Table 2 Haemodynamic quantities, known from experimental work, which have been used to assess the model narameters Quantity

Value

Source

( QI.CA)GR

3.33 ml s-’

See reference 45

Pro&)

Time-dependent driving force

Fourier expansion33

Time-dependent driving force

Fourier expansion33

5 mmHg

See reference 45

0.2-0.3 (dimensionless)

See references 28

PF

and P@

PRA ( QU&SR-( Q,_dcso ( QL&SR

Pvenlsrunder CSO Pven~d,P under Cm

Envelope shape

See references 12-14

Envelope shape

See references 12-14

( V,“),,,

15-30ml

See references 32. 23

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433

: ;I

Simulation ofmonury cinuIution: W. Schcincret al

I

-30

-29

I’

-28

I I

4

I

:

-27

I

I

: : \

i

I I

I I , I , I I I I I I I I I

I I I

E

-25

.? 9

z

I , I I I I I

I I I , I I I I I I I I I , I I \ I

-26

-24

!

I I I I I I

-23

iI

t

‘!

,’

-------~----~~~~~-~~~~-~~~~~~~~~~~~-~~~~~~~~~~~~_~_~~~~~~~_~_____~__~__~_________

IIt

I

15

’

I

16

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17

’

1 ’

I

18

’

19

I

20

’

I

’

21

I

22

’

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23

’

1 ’

24

Time

a

20

I ’

25

1 ’

26

I ’

27

1 ’

28

1 ’

29

1 ’

30

1 ’

31

1 ’

32

1 I I ’

33

34

35

Cs.1

-28

-27

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,I’ !

I \

‘\\ \

\\

I’

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\

‘.

-23

,’

‘\ --S_,

8

-22

-21

n

~____________________________________20

I”“““‘I”“““‘I

28.0

28.5 Time

b

(5)

1

‘1

__-__________---

,,‘.

‘---_

29.0

C

Time

(s)

Volume dynamics. a, Full time course of capillary volume (----, lefi scale) and venous volume (-, right scale) for coronary sinus release (15 < t < 20 s; 30 < t < 35 s) and coronary sinus occlusion (20 C t < 30 s). b, Sin le beat resolution of capillary volume (----, left scale) and venous volume -, right scale) during coronary sinus occlusion (28C t<29s. f c, Single beat resolution during normal perfusion (same symbols and scaks) figure 2

434

J. Biomed. Eng. 1990, Vol. 12, September

2 o s >

P‘ ;d 2r TP elm mw

c

I

,*--

‘--_

/’

*-,

‘x.

40’

\

I

__--

--a_

______---

---------

______----

---------

____--

(mmHg)

Pressure

------

(mmHg)

Pressure

___--

----_ --.

_’

*’

-I

/I

‘\

\ \

i I

_________________________--------------

--------.-----___._________

--------------___-__c_____________

aa--,,--_

__________I

x---,,,-

------------__________--

---------------VW______________

_______________________I______cI

----------------..______________

______________________r______________c

:a-,,__-,__

___________________-______________r

x;r,___,_-_

_____________-________________c_____

_______c_---___--------

--d-T__

_____---

------.. _--e---C

____--c-

__-____------*

--------_^_____

-------_--_-____________

-v-M__ -

--.

-..

*-.

__,-=’

_----

____----

-----------.

_---_...-__

_____-__________-________c_c

-------------___--_-_-~~_~~

_______________c____-__-_-------

-------_--._-.-________________

-a--,_

__s________________________________--------

==-------------_________________

____________________-____m._e--------

_*

_’

1

i-#

Simulationofwnmq circulation:W. Schcincr et al.

symmetry of the model, the LAD results are equal to those for the LCX branch and are not shown in the tables and figures. CapiUaryvolume. (See dashed curves in Figures ,?a-c.) During CSR., Vpx oscillates between 0.1 and 1 ml (Figures 2a and 23. The lower limit is related to the inverse power of VP’ and the constant p in equations (6b) and (7a7, in that increasing outflow resistance for an almost empty ca illary compartment prevents total emptying. With e SO, the capillaries are distended up to 3ml in diastole (upper envelope in Figure 2a). A proportion of this volume results from arterial input and the rest is made up by retrograde flow from the venous compartment due to elevated venous pressure (Figure 4b). During systole, the major portion of capillary volume is squeezed out despite CSO. However, the residual systolic volume (lower envelope) increases to 0.8ml within the LCX bed near the end of the coronary sinus occlusion. As a result, single beat resolutions during CSR and CSO (Figures 2b and 2~) show similar basic forms of Vz, except for absolute height an x a pronounced flat trough in Figure 2c corresponding to maximum emptying during systolic squeezing. Venousvolume. (See solid curves in Figures 2a-c.) V,,, oscillates between 21 and 22ml during CSR (Figures 2a and 2c), the small amplitude being due to easy drainage via the coronary sinus (small R_+RA). During CSO, V,, is distended b up to 29ml in systole (upper envelope F@re 2a), J espite the drainage via R,,. In diastole, part of the volume penetrates retrogradely towards the capillaries (see Figure 4b) and a smaller amount drains via R,. However, diastolic backflow ceases rapid1 as soon as the capillaries are simultaneously filled B om the arterial side. Hence, the residual volume remains severely elevated (lower envelope Figure 2a) when compared with CSR. In single beat resolution during CSR (Figure 2c), V,,, is seen to increase during early systole, followed by a quasi-ex onential decline in diastole (free drainage to RI ). Conversely, during CSO (Figure 2b), V,,, runs parallel to myocardial squeezing, since backflow to the capillaries cannot start until after the end of contraction. Capillarypressure. (See dashed curves in Figures 3a-c.) P$$ runs mainly parallel to myocardial squeezing, P%“(t). Only the diastolic envelope (Figure 3a) is slightly elevated during CSO, owing to back pressure from the venous compartment. V&us pressure. (See solid curves in Figures 3a-c.) Starting from small oscillations above atrial values, P,,, shows a very characteristic form of rise during CSO, which is well known from canine and human studies 12-24.In particular, the increasing gap between systolic and diastolic envelopes served as a crucial experimental fact upon which the modelling of the venous corn liance was based. During CSO, P,, slightly lags E ehind Pz (Figure 3b). Flow j?om arteries to capillaries. (See dashed curves in Figures 4a-c.) Compared with CSR conditions, Q z!&, declines on average during CSO (Figure 4a) as 436

J. Biomed. Eng. 1990, Vol. 12, September

observed in canine studies28. Since LAD and LCX are modelled s metrically, the main left coronary arterial flow $f&A = Q~ZLp + Q#&) is simply double the values shown in Figure 4. Early systolic capillary compression drastically curbs arterial inflow to zero (the dashed lower reference line in Figure 4c). During late systole, there is again little forward flow, which increases to a pronounced maximum in diastole. When compared with ex erimental measurements 2g these patterns agree we1P with flow curves found in basodilated beds and even the notch after early systolic decline is reproduced. Note that the diastolic maxima are approximately double the mean arterial inflow to compensate for systolic flow impedance. During CSO, there is even some arterial systolic backflow, since more ca illary volume is available to be squeezed out an B elevated venous pressure impedes forward egress (see Fijye 2). However, the phasic shape of flow remains more or less unaffected by CSO. Flow from capillaries to the venous compartment. (See solid curves in Figures Ba-c.) Little forward flow in diastole and a pronounced maximum during early s stole characterize the phasic form of Qz+,, under 6 SR conditions (Figure 4~). CSO, however, induces a very significant change: during diastole there is a massive backflow to the capillaries (the dashed upper reference line in Figure 4b), forming the upper envelope of V”$ in Figure 2a. This increased diastolic capillary volume is in turn squeezed out during systole, producing the massive peak of Q$T_,,, seen in F&we 4b. Impact of venous compliance on the rise of venous pressure For normal perfusion (no coronary sinus occlusion), the exact choice of venous compliance is not essential to reproduce the arterial haemodynamics found by measurement2931. For coronary sinus occlusion conditions, however, the appropriate modelling of venous compliance becomes crucial for the correct reproduction of the shape of rise in venous pressure, as can be seen from the following simulation results. Figure 5a shows the full time courses of P,,, under the assumption of constant venous corn liance. Setting u = Oml-’ in equations (4) and (5) ma!es C,, independent of I!,,. Integration was then carried out for x = 0.2, 0.5 and l.OmlmmHg-’ (parameter study). It is obvious that the typical envelope form, as found in measurement studies, does not arise for any selections of x. The diastolic envelope rises slower than but linearly with systoles, and the progressively increasing p fails to turn up with x = constant. Moreover, 8 e most realistic form of envelo e (represented by the uppermost dashed curve in P igwe Sa), corresponding to the very ‘stiff’ choice of x = 0.2 mlmmHg-I, yields a venous volume (lowest dashed curve in Figure 5b) far too small when comared with experimental findings=. On the other Rand, a larger compliance (x = 1 mlrnmHg_‘), while yielding reasonable volumes, produces unrealistic ressure levels (solid curves in Figures 5a and b). !h erefore, a volume-dependent form of venous

Simulation of conmaycircuhtian:W. Schtiucr et al.

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Flow dynamics. a, Full time course of flow from the coronary arteries to the capillaries (----, Q”” and of flow from the capillaries to the venous compartment (-e occlusion/release pattern is the same as in Figure 2. b, Single beat resolution during coronary sinus occlusion. c, Single beat resolution during coronary sinus release

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J. Biomed. Eng. 1990, Vol. 12, September

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study for constant venous compliance. O, Venous pressure dynamics for CT= Oml-’ and ,y = 0.2 (----), ). b, Venous volumes corresponding to the pressures shown in a

J. Biomed. Eng. 1990, Vol. 12, September

I ’ 33

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Figure 6 Parameter study for myocardial squeezing. a, Venous pressure dynamics for y,,, = 0 (----), y=O.5 (......) and y=O.75 (-default parameter set). b, LCX flow from capillaries to veins under coronary sinus occlusion. The symbols the same as in a c, h for b,but W& patent coronary sinus

J. Biomed. Eng. 1990, Vol. 12, September

439

Simukationofwrona’y circalutioa: W. Schincr et al.

and is also suggested corn liance proved mandato by tl! e following ph siologic ;r arguments. First, during systoTic contraction, the major portion of capillary volume is squeezed into the venous compartment (volume increment Au,,,,,~ in each beat, see Figure2). with the coronary sinus obstructed, only part of that increment can leave the venous bed again, either via QeX(mainly during systole) or retrograde1 into the capillaries (only during diastole). There rore, the residual diastolic venous volume increases as CSO persists (rising diastolic envelo e ZQ,,& and concomitantly PvenJdia rises (this would 1 e the case even for constant C,,,). However, the pressure increase in each beat, Pvenlsys - PvenIaia, following the systolic input of Au,,~.~ increases with VVen,dia if the compliance declines with increasing V,,. Physiologically speaking, with a volume-dependent compliance, the same Au,~+~ induces a ‘more than proportional’ rise in Pv, if there is an elevated volume preload V,,, ldia (see Appendix A for a mathematical treatment). Second, due to their collapsible nature10~32,veins readily accommodate increased volume up to a certain point, above which their elastic reaction rapidly increases (compliance diminishes). Thus, C,,, is a declining function of IT,,,. However, the exact functional form of equations (4) and (5) is arbitrary to some extent. Furthermore, it is interesting to compare the venous compliance derived from our model with totally inde endent experimental findings. First, experimen tap data also suggest some h erbolic or ex onential decline of compliance wixp increasing voPume22~26~33.Numerical values refer to venousspecific compliance rather than compliance itself, and range around 0.01 mmHg_’ at P = 10 mmHg25. As a check, venous-s ecific compliance may be calculated from equation Q 5) for arbitrary pressures:

c”,“Ke3= VW”

1 P”,“(1+ u V-m)

(12)

Inserting the parameters xv,,, (T and VO,,, from Table 7 and combining equations (4) and (12), venous-specific compliance from our model is found to be 0.013 mmHg-l at P,, = 10 mmHg, which is in perfect agreement with the experimental value. Note that the non-linear equation (4) has to be solved iteratively for V,, in order to evaluate equation (12). Parameter study for myocardial squeezing The impact of capillary squeezing on pulsatile pressures and flows is anal sed further in a ammeter study using the defaurt parameter set Pincluding volume-dependent venous compliance) with varied squeezing pressures. Figure 6a shows P,,, for = 0 (dashed curve), 0.5 (dotted curve) and 0.75 iy”“Fd so 1 curve). However, the reaction of pulsatile flow to the venous compartment (Figum 6b and 6~) is more interesting. During CSR and without s ueezing (P”,“z” = ‘yno,,,, = O), Q@Ln is steady at a leveP almost double the mean flow for yn,, = 0.75. In this case, it is the static resistance alone which determines flow (see Table 4). With increasing squeezing action, the peak flow occurs earlier in systole and declines more rapidly, with a concomitant decrease in mean flow

440

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Table 3 Haemodynamic quantities calculated from the model. ‘Mean (CSR)’ gives normal perfusion estimates and ‘mean (GO) refers to the average taken over 20s of coronary sinus occlusion. In the symmetric case (i.e. y 2 = 1 and RE = 0) calculated quantities are eoual for the LCX and LAD comuartments. Since the heart rate is 6Odpm, flow values in mls-’ also ‘give single beat volumes. Forward and backward components of flow were averaged separately (backward flows are denoted as negative) Unit

Mean (CSR)

Mean (CSO)

ml s-’ (ml min-‘)

3.63 (2 18)

2.97 (178)

mls-‘(mlmin-‘)

1.82 (109)

1.5 1 (90.4)

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ml SK’(mlmin-‘)

= 0.0

-0.0217

Forward Qz_yen

mls-’ (mlmin-‘)

1.83 (109.5)

3.03 (182.7)

ml s-r (mlmin-‘)

0.0

- 1.54 (-92.6)

ml s-r (ml min-‘)

3.3 1 (198.8)

0.0

Quantity

Q Fo:ard

Qz;,

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Qk&..

(- 1.3)

?-W $x CBP

ml s-’ (ml min-‘)

0.35 (2 1)

2.88 (172.8)

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0.418

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(Figure 6~)~~.Under CSO conditions (Figure 6b), a pronounced forward maximum in systole and a retrograde maximum in diastole develops. Peak flows (in both directions) are about three times larger than the average flow observed during CSR (see Table 4). Mean values of haemodynamic quantities Mean values of haemodynamic quantities obtained from the model (using the default parameter set) are displayed in Table 3. For each uantity two kinds of results are given: the ‘CSR’ va4ue corres onds to a simulation without any coronary sinus occ Pusion, and the ‘CSO’ value is a mean over a very long (20s) occlusion. Furthermore, forward and backward components of flows are averaged separately. Table 4 shows some selected quantities especial1 affected by variations in myocardial squeezing. Th ese mean

Table 4 Impact of myocardial squeezing on haemodynamic quantities. A denotes differences between the mean values obtained under full squeezing (‘y._ = 0.75; cf. Tub& 3) and the means obtained without any squeezing (x_ = 0). For example, since (Forward Q”_.,_JCSR= 1.82mls-’ under full squeezing (cf. Tubk 3) but 3.18mls-’ without squeezing, A,, = - 1.36mls- . On the other hand, (Backward Q”” _+,_v~)cm = - 1.54mls-’ under full squeezing but zero without squeezing; A,-,, = - 1.54 mls-‘. Again, LAD and LCX values are equal. Note that for positive quantities ( QLca, forward flows and volumes) a negative value of A indicates a decline as a consequence of squeezing, whereas for negative quantities (backward flows) A < 0 represents an increase Unit

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values correspond to the pulsatile discussed in the previous section.

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DISCUSSION A variety of models (recently reviewed by Mates35 has been established to explain experimental results pertaining mainly to the arterial s stemg~22~25.36-3g.In the present work, we try to estab %‘sh a simple model for evaluation of those haemodynamic mechanisms relevant to the venous bed and to coronary sinus interventions. For the sake of simplicity we do not take into consideration inertial effects and compliance on the epicardial arteries, since this would introduce several additional model parameters without substantially changing the consequences for the venous bed. However, the present model also realistically predicts phasic arterial flows, even though respective measurements3* were not utilized to adjust the model parameters. This may be seen as indication of consistency. Some features of the model, such as (i) vessel stiffness increasing with distension, (ii) vessel resistance being proportional to the inverse square of intravasal volume (Poiseuille’s law) and (iii) vasoconstrictive pressure adding to myocardial squeezing pressure to yield effective intravasal pressure, are similar to the previous reports mentioned above. They differ horn other approaches” relating inflow resistance directly to PLv via a squeezing factor which summarizes the inflow impedances due to squeezing pressure and reduction in vessel diameter. Both approaches, including the model described here, seem to be more realistic than a recent study4’, which, in an electrical analogue, abruptly curbs retrograde arterial flow b the use of diode valves rather than a resistance w Zich increases smoothly as the intravasal volume declines.

Model adaption and predictions The present

simulation demonstrates that experimental data obtained during coronary sinus occlusion can provide a basis for a realistic modelling of the coronary venous bed and thus supplement previous modelling ap roaches which have focused on the arterial side o P the coronary circulation. Mathematically speaking, data obtained during CSO provide additional boundary conditions (JZ,,_,_,RA= a), to which the model can be adapted. This reduces the degrees of freedom within the model, and thereby increases reliability (see the right hand columns in Tables 7 and 2). Sensitive and controversial issues are capill compliance and the volume resulting therefrom. a!? or experimental results obtained from the arterial system41d3, it is sometimes difficult to assess which (proximal) portion of the capillary bed has been involved. This may be a major reason why experimental results show so much divergence. For example, the value of 0.035 ml mmHg_’ per 100 g LV mass at a pressure of 20 mmHg25 is almost six times smaller than C&’ = 0.2 mlmmHg-’ resulting from our model (assuming that the LCX bed comprises approximate1 1OOg of myocardial tissue=). Conversely, Wie dyerhielm26 estimates specific compliance as O.O15mmHg-’ at P=5OmmHg and 0.005mmHg-’

at P = 100 mmHg. Using the model relation: CLCX/ vy C=P

= (PF)-’

(13)

yields 50-l = 0.02 mmHg-’ and loo-’ = 0.01 mmHg-’ for specific compliance (distensibility). This agrees reasonably at Pgx = 50mmHg; however, the result from the model is twice the experimental value at 100 mmHg. Other estimates, considering intramyocardial volume changes and variations of pressure gradients25 arrive at a compliance of 0.2 mlmmHg-‘, which is identical with the model result. This agreement is remarkable, since (in the model) capillary compliance had been assessed together with venous compliance solely on the basis of venous pressure rise. For the phasic change of capillary volume during each heart beat, the model yields 0.8ml (for CSR conditions) which agrees with the literature (0.75 ml) 12. The mean value ( VkF) = 0.418 ml, however, presents the mean of the ‘movable’ or ‘squeezable’ portion of the capillary volume rather than the ‘total ca illary volume’, which is known to be about 5-10m P per 1OOg tissue23.

PHASIC REDISTRIBUTION OF BLOOD DURING cso The s ueezable part of the capillary volume, which is cruci J for myocardial nutrition and for the washout of metabolites, was found to be distended during CSO to five times its original value (2.2 ml as o posed to 0.418ml during CSR, see Table 3). The dp iastolic backflow from the venous compartment amounted to 1.5 ml per beat, which is approximately two-thirds of V &fdia(see Table 3). Thus, according to the model, phasic backflow during CSO proves to be about equal to the forward flow durin CSR (1.82 ml per beat); concomitantly, during CS 8 the forward component of flow to the veins also increases (to 3.03ml per beat). However, the balance between forward and backward corn onents results in a decline in the mean net flow Buring CSO: (Q$l,,,) = 3.03 1.54 = 1.5 ml per beat. Hence, LAD and LCX 2 x 1.5 = 3.0ml per beat during CSO in Tabb 3). Compared with CSR (3.63 mls-‘) this reduction is in agreement with experimental values 28. As a result, oscillations of LCX during CSO are similar in ma itude to the me”“z”iow durin CSR, and in each %n eat approximately two-thir li s of the capillary volume is exchanged with the venous compartment. It is suggested that this enforced exchange of blood between capillaries and veins equalizes patholo ‘Cal accumulations of toxic metabolites (the ‘was a out effect’) and probably plays the key role in the therapeutic application of pressure controlled intermittent coronary sinus occlusion (PICSO). As to the efficacy of this intervention, the quantity of redistributed volume seems sufficient to expect an increased washout effect. However, the present simulation assumes active myocardial contraction within the LCX and LAD beds and therefore only applies to the use of PICSO as an interventional assistance to overcome global moderate ischaemia with more or less intact contractility (e.g. in the reperfusion phase during surgery following cardiac arrest). In this case, nutrition is not the crucial issue, since it is provided

Q

J. Biomed. Eng. 1990, Vol. 12, September

441

Simukatios

ofwronarycirculation: W.Schemeret al.

by the arterial system, although it may be slightly reduced to ether with the depression of ( QLCJcsc (see Table 4. To study the haemodynamic effect of coronary sinus interventions in an acute LAD infarction setting (severe regional ischaemia), the model calculations need to be carried out asymmetrically between the LCX and LAD beds (introducing RE>O and rE < 1). Further simulations using this kind of model seem desirable in order to investigate the impact of parameter variations on various coronary haemodynamic quantities. The changes in myocardial redistribution of blood during a reduced regional contraction and the ap lications to other coronary sinus interventions sueK as synchronized retroperfusion (SRP) or combinations of SRP and PICSO, may also be investigated. Mathematically, these interventions can be modelled by changing the boundary conditions for venous outflow (e.g. by imposing retrograde venous flow as an additional driving force during SRP). ACKNOWLEDGEMENT The calculations for this work were performed on an IBM4381 mainframe at the Institute for Medical Computer Sciences, Universi of Vienna. The simulation was coded in FOR TRZN , and SAS was used for the graphics. We gratefully acknowledge the s stem support given by Christian Reichetzeder and x e assistance of Christine Nanninga, MD.

12.

13.

14.

15.

16. 17.

18.

19.

20.

21.

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22.

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Simulationofwmnay tircuhtion: W 32. Liidinghausen M. Nomenclature and distribution pattern of cardiac veins in man. In: Mohl W, Faxon D, Wolner E, eds. Clinics of CSI. Roceedings of the 2nd In&national

Symposium on Myoeardial l+ot&ion 33.

34.

35. 36.

37.

38.

39.

40.

41.

42.

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via the Coronary

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Appendix: Effectof volume-dependentcompliance The following is a mathematical treatment of the venous pressure increase AP following a systolic volume increment A I? Let the venous compartment be distended (due to CSO) up to a diastolic residual volume V,. In the case of a constant compliance C the pressure increase is proportional to A V, regardless of V,: Ahp=P]T-P[,ia=$(v,+AV-

a)=?

(Al)

If C is a decreasing function of I’, i.e. C(vd+Av)<

C(K)

(M)

for all V, then the increase in pressure not only depends on A V but also on V,: I$+AV “=

C(&+AV)

--

v, C(v,)

1

=C(v,+AV) The expression in parentheses is always larger than zero due to equation (A!T!).Thus, in addition to the first term which is proportional to AV, the second term adds another increase in pressure as V, becomes larger.

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