AME RICAN JOLZLNAL OF Vol. 228, No. 1, January

PI~YSIOL~GY

1975.

P7intd

in U.S.A.

A three-element papillary muscle STANTON

model

describes

A. GLANTZ

CE velocity vs. time; isometric quick release; isotonic release; isotonic quick release; mathematical model; myocardium; parallel element;

series-elastic

cat

elasticity

STANTON A. A three-element model describes excised cat GLANTZ, 1975.papillary muscle elasticity. Am. J. Physiol. 228( 1) : 284-294. The three-element model for skeletal muscle has been widely applied to cardiac muscle. It consists of an active contractile element (CE) that represents the muscle’s response to stimulation, in series with an elastic element (SE) and the CE and SE in parallel with another elastic element (PE). There have been problems in interpreting experimental data on muscle elasticity using this model. Data seem to indicate that SE force depends not only on instantaneous length, but also initial length; it is not only elastic. Recent experiments seem to indicate that the SE has time-varying properties; it is not passive. This paper formulates a three-element model in which q(x) = a[efi(X-2*) - I] governs the elastic elements, where cp = force, a, p = spring constants, x = length, and X* = rest length, which avoids these problems. The SE and PE have the same properties. (Typical values: Q m -045 g/mm2, P z 5.9 mm-lfor cat papillary at 29°C.) By accounting for the nonlinearity of the SE-PE interaction, this three-element model leads to predictions that agree with published data on excised papillary muscle’s elastic properties.

elastic

excised

Pollack et al. (14) and Noble and Else (9) explicitly raise a much more serious objection to the three-element model: they assert that the SE’s properties vary with time when the muscle is stimulated. Based on their interpretation of their experimental data, they conclude that the threeelement model does not apply to cardiac muscle. I will present a mathematical analysis which demonstrates that these and other published experimental results concerning cardiac muscle elasticity (1, 2, 5, 8-18, 20, 2 1) can be described and unified by this precisely formulated three-element model. In addition to resolving these ambiguities, this precisely formulated three-element model leads to some surprising conclusions : quick-release experiments do not measure the SE stress-strain curve, but a quantity that depends on both elastic elements; simply subtracting the PE stress-strain curve from the force-extension curve obtained in quick-release experiments does not yield the SE stress-strain curve; a preload an order of magnitude smaller than the afterload has significant effects on quick-release results; and, perhaps most surprising, the SE and PE have the same properties.

element RESULTS

(6) proposed a model for skeletal muscle which lumps the muscle’s mechanical properties into three elements (Fig. 1): an active contractile element (CE) representing the muscle’s response to stimulation and two passive elastic elements, one in series with the CE (the series-elastic element, SE) and one parallel to the CE and SE (the parallel elastic element, PE). Figure I shows one common way of visualizing this model. Both elastic elements are springs: their force depends only on their instantaneous length. (This assumption precludes using this model to analyze muscle viscoelasticity.) More recently (1-3, 5, 10-13, 15-18, 20, 2 1 ), various forms of this model have been applied to cardiac muscle. Some questions have been raised about this model’s applicability to cardiac muscle. Sonnenblick (16), Parmley and Sonnenblick (12), and others collected data on the elastic properties of excised cat papillary muscle and, while they concluded that the SE and PE obey exponential stressstrain laws, their data have been interpreted in terms of the model in Fig. 1 to show that the SE’s properties are not unique, but depend on the muscle’s initial length (16). But, if the SE is indeed a spring, by definition its force should be a function only of instantaneous length. HILL

This three-element model follows from three assumptions : I) The CE, SE, and PE are arranged geometrically as shown in Fig. 1; 2) The CE is freely extendible when unstimulated; CE velocity varies continuously with force when stimulated; 3) The SE and PE are pure springs governed by

and

where = force in SE, R?l = force in PE, PF as, P s = SE spring constants, constants, QP, P P = PE spring = SE length, XS = PE length (muscle length), XP * = SE rest length, XS * = PE rest length. XP Figure

2 shows

the model

kinematically

and

serves to

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EXCISED

CAT

PAPILLARY

MUSCLE

285

ELASTICITY

isotonic release, CE velocity vs. time (12, 16), isotonic quick release ( 1Z), and isometric quick release (9, 14). I will use the three assumptions above to analyze each experiment and show that the experimental results are consistent with this three-element model. Isotonic release (ah called isohmi~ afterloaded method). An isotonic twitch may be divided into two phases. First, the muscle contracts isometrically until it develops a force equal to the load, PA, and then it shortens isotonically. Let tA be the time the isotonic phase begins (Fig. 3). We shall call PA the afterload, though some authors (5, 11, 12, 15, 16) define the afterload as the additional load (measured above the preload used to stretch the muscle to its initial length prior to stimulation) in the muscle during the isotonic phase. They assume that the PE carries the preload and the SE carries the additional load (what they call the afterload) during shortening. This assumption ignores interactions between the SE and PE during shortening. Sonnenblick (16) set out to determine Young’s modulus for the SE using the following logic: the chain rule for differentiation applied to Young’s modulus for the SE yields

EL EMENT

CONTRACTILE

IC SERIES ELASTIC ELEMENT

i FIG.

LoAD 1. Three-element

model

for

muscle

‘r (7 I

4&p

d

The equations to be derived require two special cases of equation 4: during the isometric phase of a contraction, the muscle length is fixed so xp(t) = xp(0), and

during

the isotonic

phase, the load is constant,

and

PO) = PA = 5%(+(t) - do> + dKe(t))

(61

Since the GE is freely extendible when the muscle is at rest, experimenters must stimulate the muscle to obtain information about the SE. They have used four methods:

FIG.

3.

Isotonic

twitch

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S. A. GLANTZ

286 teraction

equation5:

of the SE and PE. Differentiate

. PO> =

-aSPScSS("P(0)--zC(t)-~S*)~(t)

during the isometric solve for $p:

phase.

(12)

equation 6 and

Differentiate

Y(tA) = PWA) + (as + %JP 13~hpw--z~w--zg*)

cep(t)

@Se

=

' a,@>

~~~xpw---zc~~)-~s*’ + CupppdBP(~Pw-rP*)

WPsf

(13)

during the isotonic phase. Evaluate equation 12 at tA- and equation 13 at t*+, then substitute them into equation II to obtain WA) +

= Psmd

( (Pp - P&dp(xP(o)-xp*)

+

(~1s

(14)

+

a~)&]

But ~p[f?~P~xp(o)-xp* the preload. obtain y(tA>

=

merits provide indirect confirmation because this conclusion leads to predictions which are consistent with the other experimental results. With the conclusion that /3s = & = & equation I4 reduces to

Substitute

-

l] = (op(Xp(0))

from

equution

= Pi

(15)

15 into equation 14 to

Thus, our three assumptions lead us to predict that Y&) should vary linearly with P&), as Sonnenblick experimently demonstrated (Fig. 4); this three-element model is compatible with his data. Thus we see that Y(tA) depends on SE and PE properties and is not simply Young’s modulus for the SE, d&dxsl tA-0 In other words, equation 10 is not correct because one cannot justify ignoring the interaction between the SE and PE. Note that if one ignored the fact that i) and $p are not evaluated at precisely the same time and simply applied the chain rule for differentiation to equation II, one would conclude that Y(t,) = -dP/d+. But, inasmuch as both i) and $p are discontinuous at t = tA, the chain rule does not apply. APPENDIX A shows that =

y(tA>

pSP(tA)

+

(a&P

+

&F&S)

+

(PP

-

PS)pi

(16)

Sonnenblick’s experiments showed that Y (tA) is independent of preload Pi. In other words, aY(t,)/aPi = 0. Differentiate equation 16: dY(t*)/dPi

= PS - pp = 0 P S = P I?

dP(tA)/dXp(tA)

(-m

Later in the paper we will need the integral Substitute equation 20 into equation 19: WA)/~Q~A)

and integrate

(17)

=

mt*>

+

(Qis

+

of equation 20.

(20

%JP

to obtain Axp = -1 1n PA + Pi + P

(18)

More recently, Parmley and Sonnenblick (12) noted that Sonnenblick may have been measuring peak rather than initial isotonic shortening velocity. We will show below that such an error merely multiplies equation 14 by a constant and, therefore, it does not affect the conclusion that ps is nearly equal to Pp* In order to describe the data with the minimum number of parameters, we conclude that & = PP = P * While only the isotonic release experiment allows us to conclude directly that /3s = Pp = P, the other three experi-

W)

(QIS + cyP> (as

+

(22)

aP)

where Pi is the constant of integration. GE velocity ZIS.time. During the isometric phase & = and the SE length change during this phase is t t &(t) dt = Axs(t) = &(t) dt s0 I0

- k, (23)

By measuring P(t) during an isometric twitch, if given Axs(t), one could cross-plot Ax&) and P(t) to obtain Axs as a function of P, the SE length-tension curve. Unfortunately, it(t) cannot be measured. If, however, one neglects the PE, equation 8 holds and equation 23 becomes t

Ax&>

= -

tA=t

&(t)dt

s0

= -

s t A=0

&h4)

(24)

p tA=t x-

I

dtA

b( tA) dt,

Sonnenblick (16) ex p erimentally determined - %p(tA) for many values of tA by observing isotonic contractions with various afterloads. (Changing the afterload changes tA and - kp(tA).) He computed Ax,(t) by graphically integrating tA=t

Ax,(t)

AFTERLOAD,

FIG. 4. ResuXts stress-strain curve; perature = 22°C.

PA

I gm 1

of an isotonic release experiment. This is not a stress-strain curves relate force and length. Tern(Data from SonnenhIick (16), Fig. 523.)

= -

s tA=o

b( tA) dt,

(25)

using his experimental values as shown in Fig. 5. The Axs vs. P he obtained by cross-plotting Axs(t) and P(t), however, depended on initial length (preload) (Fig. 6). (As with the isotonic release method, Parmley and Sonnenblick (12) may have overestimated the initial shortening velocity. We

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EXCISED

CAT

PAPILLARY

MUSCLE

287

ELASTICITY

Since the CE is freely extendible at rest, the SE must be at rest length at f = 0 when the muscle is stimulated: xc (0) = xp(0) Change

o0

TIME Of

2

WITIAL SHORTENING,tA ( msec 1

4 FORCE,

6 0 P, (gml

5. CE velocity vs. time method. Experiment proceeds as follows : 3) allow muscle to shorten isotonically against different afterloads and note time of initial shortening and initial shortening velocity; 2) compute Ax&) by measuring area under initial velocity of shortening vs. time of initial shortening; 3) measure an isometric twitch; 4) crossplot Ax&) and P(t). Axs is not simply related to XS. (After Sonnenblick (16), Fig. 4.)

-

TEMP+22

Ha

C ,4’

(20

equation 26:

In R3Pse

J

=

/3~~xpHO--zC

Ps

Rewrite

asps

equal&

w-~~*) +

+

ap

BP $P(zP(oJ--zP*)

(29)

Bp(~pw--zp*l

aPPP&

5:

Since xc(t) has the same value when isotonic shortening begins at lime tA = f as at time l during an isometric contraction, we can use equafion 30 to eliminate x&t) from equation 29 to obtain

B d

*H’

and integrate

xs*

ax,(t)

FIG.

IO-

variables

-

=-

1

ln

psp

+

p&s

PS

+ aa&

aP> +

+ aPPPe

aP(PP fipbpm--zp*)

Ps)eBp~~P~o)-ZP*)

(31)

6 0.1

1

0.3

0.4

I

0.5

I

0.6

0.6

I

13.5

1

0.7 0.8 Ax,, (mm1

Substitute

t

1

0.9

I.0

BP = P and equation 15 into equation 31;

from&=

1.2

Axs

FIG. 6. Results of CE velocity vs, time method for various initial lengths (preloads). Increasing preload shifts curves up with little change in slope as equation 32 predicts. (Data from Sonnenblick (16), Fig. SD.)

will see below that such an error changes the apparent magnitude of & but not the form of the relationship between P and A+) Based on the assumption that the PE is negligible, Sonnenblick interpreted his curves as the SE stressstrain law* His interpretation requires that the SE force be a function of both instantaneous length and initial length; this interpretation is inconsistent with the assumption that the SE is a spring. On the other hand, by consideringthe PE to be significant, we derive an equation that both uses an SE whose force only depends on instantaneous length and that describes Sonnenblick’s data. We begin by substituting equation 13 into equation 25, to obtain

= -1 1n p + (as + aP> P Pi + (a5 + ffP)

The presence of both as and ap in equation 32 indicates that the data obtained using this method reflect the interaction of the SE and PE, not the SE stress-strain law. This equation predicts that the P vs. AX s curve will vary with preload (initial length). Figure 6 shows that the experimentally determined P vs. Axs curve shifts as equation 32 predicts with little change in slope (which @ primarily determines); this

t< tR

t= tR

t > tA

P

tA=t =-

I

tA=O

a&Se

asPs &(sp@)-~&A)-~S*)

,S~(rp(o)--rC(tA)-~S*) +

apppJ3PbPoN--zP*)

l &d

dt~

t

% (26)

FIG.

7. Isotonic

quick-release

+R experiment

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288

S. A.

which cannot be solved in general. Fortunately, already shown that ps = pp = p, which allows equation 34 for x&): i+(t)

1

GLANTZ

we have us to solve

PA+ (as+ ap>

= - In as pzc P

(0 +q*1

--hP* +

(36)

aupe

If we release the muscle to afterload PA and reference afterload PA’, we can compute the difference in the quick-shortening phase, XA - xAf, using equation 36: LENGTH=

0

0.1

0.2 0.3 EXTENSION,

7.5mm 0.5 gm

WITH PRELOAD

0.4 0.5 ~3 IqImm)

1 PA + (w + w4 Ax, = XA - XA’ = - h 6 PA’ + (as + ad

0.6

8, Results of an isotonic quick-release experiment. Solid lines computed using actual preloads in equation 38 with (C~S + a~) = g and p = 7.5 mm-l; correlation coefficient, rl equals .971. (Data Parmley and Sonnenblick (12), Fig. 9.)

FIG.

were -163 from

three-element model in which ,& = Pp = P is compatible with these data. Isotonic quick release. The isotonic quick-release technique put forth by Wilkie (19) and Jewel1 and Wilkie (7) has gained wide use in studying the elastic properties of muscle. This method, illustrated in Fig. 7, proceeds as follows: the muscle is stretched to some initial length using a preload, then fixed and stimulated. The muscle is suddenly released during the isometric twitch to a constant load below the current isometric force; it shortens, rapidly at first, then more slowly. The elastic elements ad*justing to a new length (force) cause the rapid shortening, and the CE causes the slower shortening. Using cat skeletal muscle, Zajac (22) showed that after the rapid phase the muscle behaves as in a normal isotonic contraction. Finally, the extension is computed by subtracting the quick shortening obtained with an arbitrary reference afterload from the quick shortening obtained with other, generally higher, afterloads? After release and quick shortening, the muscle shortens isotonically with afterload PA according to equation 6: P* = crsCeSs(“F(t)-“C(t)-~s*)

_ 1-j +

We wish to solve this equation for x&) so that we can compute the quick shortening. Isolating xp in equution 33 yields P* +

(as + cyr) = [Ql~e--~~(sc-~s*)](e”p)~s

+ (34

[Q!pe-BPxP*]

which

(@P)

BP

is of the form A = 13xnl + Cxn2

In studies with cardiac muscle, it is common to make reference afterload equal to the preload. Therefore, P A’ = P;, the preload, in equation 37; = -1 In

Ax,

P

+ +

b

+

ad

(W

+

QIP)

the let

Equation 38 shows that, as with the other two methods, the isotonic quick-release method measures a variable that depends on both the SE and PE, not merely the SE stressstrain law.2 The result depends on the preload because the quick-release extension, A3cQ, is measured relative to the shortening with only the preload. Significantly, equation 36 predicts the shift in Parmley and Sonnenblick’s (12) isotonic quick-release curves (Fig. 8) with changing preload (initial length). The solid lines on Fig. 8 were computed using the actual preloads, and parameters (Q[~ + ap) = .I63 g, and fi = 7.5 mm-l. (Hartley’s (4) nonlinear least-squares regression technique yielded (as + ay) and p to best fit the data.) Yeatman et al. (21) conducted isotonic quick-release cxperiments, then graphically differentiated the force-extension curves they obtained to show that dP,/‘dAx,

= KPA + c

where K and C are constants independent result follows directly from our formulation.

(39) of preload. This Differentiating

2 One referee raised a point worth quoting and answering here. He said, “The author says that equation 38 shows that the isotonic quickrelease method measures the interaction between the SE and PE, not merely the SE stress-strain law . . . this is true only if one releases to Pi, the preload. This is what ParmIey and Sonnenblick did, and is the mistake that led them to conclude that the properties of the SE change with Pi. If one releases to zero (or close to it and extrapolates down to zero) one avoids this error, and it is independent of preload.” The experiment the referee proposes uses zero force as the reference afterload, PA’, in equation 37 instead of the preload. In this case, equation 37 becomes 1

(35 Axg

1 For example, Wilkie (19) uses 1.1 g as reference afterload in his original paper. His Fig. 3 shows zero strain (Le., extension) with this stress (afterload). His Fig. 2 shows a .9-mm difference in quick shortening observed with I.1 g the reference afterload and the 11.1 g afterload. This difference appears as the uppermost solid point on Fig. 3 : stress = 11.1 g, strain = .9 mm. Jewel1 and Wilkie (7) later followed the same procedure using a l-g reference afterload. Note that their Fig. 6 shows zero extension with a l-g load. Their Fig. 3 shows the raw quickrelease data.

PA Pi

(37)

=

zn

PA

+ (as

6%

+

+

ad

W>

The referee correctly asserts that the quick shortening phase, AXQ, does not depend on Pi in this experiment. One could conclude that the referee’s method yielded the SE stress-strain law if the equation above was equivalent to the SE stress-strain law, equation I. But, the equation above is not equivalent to equation -I; furthermore, the fact that this equation contains elastic constants for both the SE (crs) and PE (a~) shows that the AXQ obtained with this experiment depends on both SE and PE properties.

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EXCISED

CAT

the isotonic

which

PAPILLARY

quick-release

MUSCLE

eqz&on

289

ELASTICITY

36 implicitly,

we get

we can solve for dPA/dAxQ: dP*/‘dAx,,

= iw4

+ P&3

+ aP)

W)

Equation 41 is identical with the isotonic release equation (equation 21), as it ought to be identical. Indeed, the isotonic release can be viewed as a quick release in which PA equals the force in the muscle at the instant before release. Notice that equation 41 is independent of Pi; the computational method used to evaluate isotonic release data produces results that are independent of preload. Other authors (5, 11, 12) have differentiated quick-release force-extension curves to deduce K and C for isolated cat papillary muscles. (Edman and Nilsson (2) obtained analogous results using isolated rabbit papillary muscles.) After correcting their data by including the preload (APPENDIX B), we can use their values of K and C to compute a and p (Table 1). (Assuming as = ap = a will be justified later.) Comparing terms in equations 39 and 41 shows that P = K and a = C/K The results in Table 1 imply that a E -045 g/mm2 and below the a = P E 5.9 mm+, values that are substantially .081 g (muscle area is about 1 mmz) and P = 7.5 mm+ computed directly from equation 38 and data in Fig. 8. This difference probably arises because the muscle used to find the larger a and P is stiffer than average. (We will return to this problem when discussing the PE.) We shall, therefore, adopt the lower values for Q and /3. While inotropic interventions, such as changing stimulation frequency (1, 15), postextrasystolic stimulation (l), calcium concentration (15), catecholamines (15), theophylline (8), and digitalis glycosides (17), affect twitch magnitude and speed, they do not change K and C and therefore do not change (as + ap) or P. These results show that we are measuring passive properties of the muscle, that is, properties not directly connected with the active response to stimulation (i.e., the CE). Parmley and Sonnenblick (13) have also shown that K and C do not change in specimens taken from cats with experimentally induced hyperthyroidism, cardiac hypertrophy, or cardiac failure. Yeatman et al. (20) have shown that K and C increase with decreasing TABLE 1. The a and P far isohfed cat f~pihry from K and C

muscle computed

Source

Parmley blick Henderson Yeatman Yeatman Yeatman Parmley

and (12)* et et et et

Sonnen-

30

.048

g/mm2

6.3

mm-l

6

et al. (5) al. (21) al. (21) al. (21) al. (1 I )

29 29 29 29 30

.050 ,041 .017 .ooot .045

g/mm2 g/mm2 g/mm2

5.5 6.0 6.1 6.0 4.8

mm-r mm-1 mm-1 mm+ mm-1

14 11 11 11 1

g/mm2

* Not the same muscle used to compute a and p directly using t C is very small and had to be read off small graphs ; eguat ion 35. hence, this result is probably due to experimental and numerical errors.

xP

3

AX, 1 tR

I FIG.

stretch,

t

9. Isometric quick-release negative for a reIease.

AXP

experiment.

I

’

”

is

positive

T

for

a

‘I

1 B Ax, (mm)

Y

oi 0 MUSCLE

LENGTH

CHANGE,

&,hn)

705 -I

I FORCE

BEFORE RELEASE, (dyne 1 16’)

P,

FIG. 10. Results of an isometric quick-release experiment. Solid lines were computed using equation 49 with (a~ + a~) = 488 dyn and p = 3.84 mm-l; r = .998. (These values were selected to obtain best least-squares fit to data.) Value of P- in panel B corresponds to value of P, with zero length change in Panel A. Panel A presents data as originally published. Note that if one believes that this experiment measures SE stress-strain curve, one would conclude that SE’s properties vary with time. Temperature = 23°C. Original length = 6.7 mm. (Data from PoHack et al. (14), Fig. 3.)

temperature-the muscle gets stiffer. This three-element model in which ps = & = ,6 is compatible with all these observations. Isometric quick release. In an isotonic quick release, one suddenly changes the force and observes muscle shortening; Pollack et al. (14) and Noble and Else (9) reversed this procedure by suddenly changing muscle length and observing the change in force (Fig. 9). They obtained forceextension curves which depended on when (during the twitch) they released3 the muscle (Figs. 164 and 11A). With some analysis they concluded that the SE’s properties changed with time (i.e., that SE stiffness at a given force changed with time after stimulus). They further concluded that the three-element model containing a passive SE was 3 Pollack et al. (14) also present data obtained by stretching the muscle (Fig. 6 of ref. 14), but these data are inconsistent with release data for the same muscle (Fig. 3 of ref. 14). Since both figures present data for the same muscle, the values of force at a give muscle length should correspond. Close examination of Figs. 3 and 6 (14), however, shows that this is not the case, even for the diastolic curve. Therefore, I did not treat the data presented in Fig. 6.

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290

S. A. GLANTZ I

1

1

I

1

I

1 tit

I

I

1 .-

J

%

(meeo)

-05.

1

f’

2 L MUSCLE

LENGTH

00.0 0.2 FORCE

CHANGE,~Qnm)

FIG. 11. Results of an isometric puted using equation 49 with (ars fr = .996. Noble fl = 4.73 mm-l; force by value of twitch tension at RF; this procedure has no effect See caption to Fig. 10 for additional

and Else (9),

0.4 0.6 BEFORE

0.8 1.0 RELEASE, P,(RF)

quick release. Solid lines were comcyp) = .I1 7 relative force (RF) and and Else normalized the measured 330 ms and plotted their results as on analysis. Temperature = 26OC. comments. (Data from Noble

Fig. 3.)

inadequate to explain their data. I will now show that this three-element model, in which the SE’s properties are constant, is adequate. Suppose we suddenly stretch the muscle by Axp at c = tR; then xp(txt+>

= x&t-)

+ Axp

Remember that ps = & = p, substitute into equation 4, and evaluate the result at +* tR

=

p-

=

~s[&2P(++zC~~R

2 and 3 and =

=

>-%*)

-

1] (4-J)

+

p(fR+)

=

p+

c

ap[&%‘(~R-)-~P*~

+h’@R+

-

-+R+)--,S*)

-

1-j

11 (44)

+

,,[&%‘(tR+)--zp*)

-

11

But assumption 2 requires that stretching (or releasing) the muscle not create a discontinuity in xc (see APPENDIX A for proof), i.e., X&R) = x&t&) = xc (tR), and, since the twitch is isometric except at t = X&R) = ~~(0). Substitute these results and equation 42 into equations 43 and 44:

tR,

P- = +fi

(xP(O)-“C(tR)-~s”) +

-

1-j

,&s(zPw--3;P*~

P, = as [ t?C~(X~(O)+AXP-X:C:(~R)-XS*)

_

-

1-J

-

I]

(45)

l] (46)

+

Equation

DISCUSSION

Magnitude of @load is important. This three-element model of muscle described by the three assumptions at the beginning of this paper is adequate to predict and explain the available experimental data on papillary muscle’s elastic properties. The difficulties in interpretation noted above follow from the assumption that the experimental procedures measure the SE’s stress-strain curve; none of these methods directly measures the SE stress-strain curve* Furthermore, simply subtracting either the preload or PE stress-strain curve from the force-extension curves obtained using these methods does not yield the SE stress-strain curve because the experimentally determined extension depends on both the preload and afterload (cf. equations 22, 32, and 38) . It is common to make the preload much smaller than peak isometric twitch tension, and on this ground, neglect it. Equations 22, 32, and 38, on the other hand, show that in order to justify neglecting the preload one must be able to argue that Pi is small relative to the term to which it is added: (as + ap). Since both Pi and (cys + ap) are of the one cannot neglect Pi. For example, order of .I g/mm2, changing the preload from . I to .3 g/mm2 will approximately halve the argument of the logarithm in equations 22, 32, and 38. It makes no difference that Pi is much smaller than peak tension+ E@t of overestimating initial shortening velocity. The first three methods predict exactly the same force-extension curve (equations 22, 32, and 38. Figure 12 shows, on the other hand, that the CE vs. time and isotonic release curves do not coincide with the isotonic quick-release curve. Parmley and Sonnenblick (12) assert th2.t this difference is

tequation tR-t (42)

l

P&-)

for a fixed stretch, A+ (a release is a negative stretch), the force immediately after the stretch, P+, should vary linearly with the force immediately before the stretch, P-. Figures 1OB and 1lB show that the data exhibit this property All the solid lines on Figs. 10 and 11 were computed using equation 49 with (as + Q’~) and /? selected to minimize mean squared error ; all the solid lines in each figure were computed using the actual value of axp and the same constant values (as + ap) and 6. This three-element model in which with these data. In fact, we must P = P - p is compatible h:ve p,’ 1 pp = fi in order to derive equation 49 from equations 43 and 44.

CYPIGBhP(0)+hzP--zP*)

46 yields

p+ =

Io .

&mP

pP~~)-ww-~a*~

_

l]

6 MUSCLES AVERAGE LENGTH PRELOAD 0.5gm \

6.4 m m

3 2

(47) +

P+ =

(.&pP

pP(oJ-zP*J

e~AzPtag[ea("P(0)-~C(tR)-"Sf)

-

_

1-j

&ii [iii#

I]

I (48)

+

cY,[e~(xp~o~-xp*~

-

l]}

+

(as

+ -

But the term in braces equation 45, so P, In other

the

(%

+

4

in Equation 48 is P- according

= eanxPP- + words,

ap)efiAxp

(~1~ + ,,)(ebAxP

three-element

model

-

1) predicts

0 0

to

(491 that

FIG. isotonic CWJ~S

amount.

’

’ 2

’

’ ’ ’ 4 6 AFTERLOAO,

’

’ 1 1 1 ’ 8 IO I2 g(grn)

12. Comparison of results of isotonic quick release (curut~ I), release (GWW Z)? and CE velocity vs. time (curve 3), Note that 2 and

(Data

3 are shifted from Parmley

above and

quick-release Sonnenblick

curve by about (12), Fig. 3.)

same

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EXCISED

CAT

PAPILLARY

MUSCLE

291

ELASTICITY

As with the isotonic release method, overestimating -%&) does not affect the form of the force-extension curves. Since K > 1, the isotonic release and CE velocity vs. time curves ought to be identically shifted above the isotonic quick-release curve; Fig. 9 shows that this is approximately the case. Furthermore, equations 53 and 55 predict that the isotonic release and CE velocity vs. time curves ought to be shifted above the isotonic quick release curve by exactly a factor of K. To see how well this prediction reflects reality, define the ratios 5

-0

IO

AFTERLOAD FIG.

Fig. frdm mm

Azi

15 pi

tgm)

from mean values 13. K’, pp’, and ~8’. pr’ and pg’ computed 12 (average muscle length 6.4 mm with .5 g preload). K’ computed data in Parmley and Sonnenblick (12), Fig. 8 (muscle length with .5-g preload).

in 8

K’

=

K

-

that allow Normalize

I If the assumption would have

us K

=

-&@A+)

--1 P(t*-)=

‘Y(tA)

-

K

K

-?&A+)

that K’

XPW

A

=

Pi

(57)

xl49

Figure 13 shows that K’ stays approximately constant, implying tha t our assu mption is, for the purposes of this arg ument, reasonable. In light of equation 50, the isotonic release will predict

w>

where i refers to isotonic release (P) or CE velocity vs. time (S) extensions, respectively. Normalize the p’s by muscle length with .5 g preload: Pi

= constant.

Parmley and Sonnenblick (12) present data to check the validity of assuming K constant. by muscle length with a .5-g preload :

(56) Ax,

due to experimentally overestimating -&(t*) at the onset of isotonic shortening. Essentially, they conclude that they were measuring the peak shortening velocity rather than the initial shortening velocity (the quick-rise component) which occurred a few hundredths of a second earlier (Fig. 3). My analysis confirms their argument. Furthermore, overestimating -$&A) does not disturb the\ arguments presented earlier in this paper. Let - & be the measured (peak) shortening velocity. Suppose it is related to the actual (quick rise) initial shortening velocity by K

=

(52)

K

is constant

were exactly

= pp’ = ps’ = constant

correct,

we (58)

Figure 13 shows that, while equation 58 does not hold, K’, pp’, and ps’ all fall in the same range. This situation adequately explains the effects of overestimating -a&), given the simplicity of our initial assumption (equation 50). Why isometric quick-release method yields fordength curues that change ~~,ith time. Figures lOA and 1 IA indicate that the force after isometric quick release depends on the release time, tR. Rather than resulting from a time-varying SE, this observation follows from the fact that the SE is not a linear elastic element. Since SE force increases exponentially with length (equation 1), a fixed length change, Axp, causes different force changes when the SE is at different lengths at TV’. During an isometric twitch the CE changes SE length. When the same Axp is applied at different times, the SE lengths (at tRm)are different; therefore, the change in force is different. (Since the releases occur during isometric twitches, PE length remains constant [equal to +(O)] until tR, so its contribution to the force change upon release de-

Notice that overestimating the initial shortening velocity changes Y (tA) by a constant factor and therefore does not affect the argument based on equation 14 that Ps = pp = p. Using equations 19 through 22 exactly as before, A?i, = 2 ln P

PA + b

+ CllP)

pi

+

+

(@

Q(P)

-kP(tA) by a constant factor changes the Overestimating apparent value of @, but not the form of the force-extension curves. Similarly, for CE velocity vs. time, we compute

A& =-st

ip(t)

dt =

--K

0

and using

gp dt

0

equations 25 through A&

ts

32 exactly

as before,

= !f ln PA + (as + cyP> 6

Pi

+

(a*

+

aP)

8

(54)

LENGTI-I

FIG-

(55)

papillary others. Parrnley Taylor

9

IO

(mm)

14.

Length-tension curves obtained from unstimulated cat muscles. Note that muxle described by curue I is stiffer than (Data sources : ParmIey and Sonnenblick (12), Fig. 10 (curve I) ; et al. (10) Fig. 3 ( curve 2) ; Parmley et al. (11) Fig. 2 (curve 3) ; (18), Fig. 3 ( curtje 4) J Temperature = 29-30°C,

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292

S. A.

GLAN?‘%

and Ax,

Thus, for a given muscle, equations 60 and 61 should, with the same values of a and ,B, describe the passive extension and isotonic quick-release force-extension curves, respectively. Figure 15 shows that this is the case. Pollack et al. (14) present the passive force-extension curve for the muscle they use in the isometric quick-release experiments described in Fig. 10. If we assume as = ap = a, equation 49 becomes

0 PE

(unstimuloted muscle) WICK RELEASE

l

TEMP

30

MUSCLE 0.4 mm MUSCLE I.1 mm2

C LENGTH AREA

i

= =

I i

2

6

P,

FORCE,P(gm) FIG. 15. Length-tension curves obtained by stretching unstimulated muscle and by doing isotonic quick-release experiments. Solid lines were computed using equations 60 and 61 with a = .062 g (056 g/mm2), /3 = .37/% muscle length = 4.4 mm-l (these values were selected to obtain the best least-squares fit to the data), and the actual experiment; r = ,996. (Data from preload = .2 g for quick-release Parmley et al, (1 X) Fig. 2, except value of preload which is by personal communication with Parmley.)

pends only on A+) As iR increases (up to the time of maximum developed force), SE length at tR+- increases and therefore the SE moves higher onto the steeper part of its exponential force-length curve. Thus, as tR increases a constant A+ yields both a greater force after release, P+ (because the SE is higher on its exponential force-length curve), and a greater change in force (because the curve is steeper) m Par&l elastic element. Thus far we have deduced elastic properties using stimulated muscle; now let us turn to unstimulated muscle. Since the CE is freely extendible at rest, we will see just the PE. Figure 14 presents four length-tension curves for isolated cat papillary muscle. Their exponential shape confirms our assumption that force varies exponentially with extension (equations I and 2). The curve Parmley and Sonnenblick (12) obtained represents the muscle that was used to compute a and P directly using isotonic quick-release data and equation 38. Note that this muscle is stiffer than the others; hence, we can conclude that the Q and P for this muscle are higher than average and that we are justified in accepting the lower values (ff ==: .045 g/mm2, P Z 5.9 mm-l) as typical of muscle at 29°C. Parmley et al. (11) present isotonic quick-release and passive extension (i.e., when the muscle is unstimulated) curves for the same muscle. Using equation 2, we ought to be able to describe the passive extension curve with xp -

xp

*

1 p + aP = - ln ----P w?

and using equation 36 we should tonic quick-release curve with A+

1

PA

p

pi

= --In

be able to describe

+ +

(a3

+

aP)

(as

+

(xP>

-

XP

*

= -1.In Ap-t-a P

a

= eBAxpP- + 2a(efiAxp -

1)

(62)

If the three-element model is valid, we ought to be able to describe the passive curve with equation 60 and the values of a = >4(as + 01~) and P derived in Fig. 10. Figure 16 shows that this is the case. In the first case we assumed as = ap in order to get equations 60 and 61; in the second case we assumed ~1~ = ap in order to get equations 49 and 60. In both cases assuming = cyp led to mathematical expressions which described ffS results obtained from different experiments. Therefore, = ap = a is a reasonable assumption. Furthermore, the % ability to describe with the same constants in different equations both actively and passively collected data indicates that this three-element model describes the data. Voight form of three-element model. Many authors (e.g., 11, 14, 20) call the three-element model arranged as in Fig. 1 the Maxwell form of the Hill model; they often discuss another arrangement, the Voight form, in which the SE is attached in series to both the CE and PE. There has been some controversy over which of these forms applies to cardiac muscle. Fung (3) showed that if one assumes a viscoelastic PE and ‘

Substitute

equation

for unstimulated

differentiation A2 for 2~:

with

$4

&, = -

PS’ +

to the function’s

to obtain

AI1

(Am

= &

= xc@+)

=

equation

lim

-

xc(t-)

(AXC/AXP)

Asp-*0

= 0

Wdxp

Iim

=

into equation All

A19

m-) = --

-

iP(tA+)

=

= 0

Y(tA)

=

d[XP(tA)

to obtain

PS’

+

PP’

(A-w

can

so we

use

equation

dP(tA)

Y(t*)

-

&A+)]

+

~P’[XP(th+)l

w(~*+>l~C~~*+)

-

-

}

dtA)]

+

the total

APPENDIX PRELOAD

Since xp(t) is is continuous. and so Xc is

‘PP’!XP(tA)!

64-w

= ~

differential

(A@

(aP,‘axc)dxc

+

= aP/aqa

+

@P/&Q)

6-w

- (dxc,‘dxp)

equations

A9

and AI0

dP/dxp By construction

Experiments

My analysis show

(Fig.

using

=

into equation

= 408’ + PPl - d

(equation

equation

C FOR

TABLE

Pi

1

= K*PA* 41)

cm

of afterload

(5, 1.1, 12, 21) show

+ C*

and corresponding = KPn

BI into equntion dPA/dAxQ

Compare Table 1.

equations

to obtain

(d-w/dxp)

+

the first definition

predicts

w4 (Am

A8

PA*

dPa/dAq Substitute

= -cps’

AND

(B-3 experiments

-+ C

B3 and

BZ to obtain

= l?PP

+- cc*

B4 to obtain K = K*

(AW

c = c*

(14) W)

(A@

4:

aP/dxc

K

Let K* and C* refer to values obtained in experiments in which afterload is defined as total force less preload. Let K and C refer to the values obtained in experiments in which afterload is defined as total force in the muscle during the isotonic phase. Let PA* be the force referred to in the first definition and PA be the force in the second.

dPA”/dAxQ

aP/axp = psf + +d Substitute

TO

(4

so

equation

CORRECTION

of P is

dP = @P/axp)dxp dP/dxlJ

B

PA

p = P(xc, XP) By definition,

A20

(A4

4 as

equation

(Am

(A3)

XC

d[xP@A+)

can write

# # a

&A-)h(tA’)

’ (cPS’bP(tA+)

=

is

(Am

(o/Axp)

Axp+O

when the muscle is stimulated. t = fA when the muscle is stimulated, with equation A5 to obtain

CPP’

In an isotonic contraction xp(t) and P(t) are continuous. continuous, pp(t) is continuous and cp&) = P(f) - (up Finally, ic is a continuous function of PS by assumption, continuous at t = & and equation A4 becolnes

From

(Am

dXP@A) dXP(tA-)

We

1

muscle.

Thus.

WA)

1 =

2) If the muscle is stimulated, suppose Ax8 = 0 at linae t. Then ps continuous at t. Since 2c is a continuous function of pa by assumption, ;irc is continuous at t, i.e., Axe = xc@-) - xc(t+) = 0. Substitute A xs = 0 and Axe = 0 into equation A13 to obtain Axp = 0. But Axp 0 by hypothesis, so we have a contradiction and AXS # 0. Since Ax8 0, m(t+> # m(P) and &(t-) # ic(t+). But since we only have simple discontinuity in Xc, xc(t) is continuous at time t:

LW

tA

respect

lim Asp+0

into equation

AI6

dxc/dxr

+f >

=

so

to time yields

rPF%,

of the derivative

(A~c/Axp)

dP/dxv

t < tA +

lim Asp+0

Substitute -

the definition

=

Axe

respect

+(t) = -&&, 0 = ps’(ip

0 and we have

20

5 and 6 with

equations

Let Axr+

-

KVi)

the preload

w corrections

for WJ)

-

KV,

@a

2) APPENDIX

Suppose at time t we suddenly change xp by Axp; AXS be the corresponding changes in xc and x8, Axe

+ Axs

= Axp

if we let AXC

and (A 13)

There are two cases: I) If the muscle is unstimulated pps = 0 by definition. Since qps is a continuous function of xg and since Aqg = 0, AXS = 0 and equation A13 becomes Axe

= Axp

LIST

% ai PTPj K Kf Pi

Pj’ PPj c K

n Axc/Axp

=

I

(Am

pi pj

OF

C NOTATWN Elastic constant Elastic constant Ratio defined by equation Ratio defined by equation Ratio defined by equation Ratio defined by equation Force Constant in equation 39 Constant in equation 39 Number of samples Muscle force

50 51 56 57

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294

S. A.

t, t, j Xi * #j

Axj Y(tA) Szcbscripts A, A’ B c * b Q R s Superscripts hr +

-.

‘I’lme Length Rest length Change in length Ratio defined by equalion

11

Afterload, beginning of isotonic End of isotonic phase Contractile element Preload Parallel elastic element Isotonic quick release Release Series-elastic element Observed Before After

phase

value

I thank Professors Gene F. Franklin and Thomas R. Kane and Dr. Michael R. Headrick for their thoughtful criticism and encouragement. I thank Drs. William W. Parmley and Alan J. Brady for reviewing a preliminary manuscript. I also thank Stephen S. Ashley for copyreading the manuscript and Judy Clark for typing it. This investigation was supported in part by Hearth Services Demonstration Grant HS 00146-03 from the Health Services and Mental Health Administration. Manuscript revision was supported by National Institutes of Health Grant 1POl HL 15833-01.

GLANTZ

These results were presented in abstract form at the 26th Annual Conference on Engineering in Biology and Medicine, September 3OOctober 4, 1973. Since this paper was submitted, Brady (European J, Curdiol. l/2 : 193, 1973) CCmeasured muscle stiffness by the application of small ( < 0.5y0) rapid stretch and release length changes to the muscle at various times. , . . The ratio of force change (AP) to the length change [AXP in my notation] . . . gives the stiffness.” This “perturbation method” produced a linear stiffness vs. force curve independent of preload (except at the highest preload tested). My model is consistent with these observations. One can use equation 49 (with crs = a!p = cu) to predict that stiffness measured this way should be (A?'/ lAxpI ) = [sinh pAxp/Axp (P -j- 2a)], a straight line independent of preload. I also predict Brady’s observation that “the perturbation method gives a stiffer appearing [muscle] than quick-release methods.” For quick releases, (AP/ [AxpI ) = (efl*‘p -l/Axl~) (P + 2cy). Since G@*~P - 1 < sinh PAXP for all AXP < 0 (i-e., for all releases), we predict that the slope of the perturbation-measured stiffness curve should exceed that obtained with quick releases. Meiss and Sonnenblick (Am. J. Physiol. 226: 1370, 1974) measured dP/dxp using quick stretches and releases and showed that it varied linearly with force, independent of preload. Equation AH shows that so their experiment provides new data that directly dP/dxp = Y(t*), lead dicts

to the conclusion their observation

Received

for

publication

that that

@s = pp dP/dxlJ

30 April

= p. In addition, this = BP +- 24 (equation

model 21).

pre-

1973.

REFERENCES

1. ABBOTT,

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