a bond graph model of chemo-mechanical transduction in the mammalian left ventricle

A bond graph model of chemo-mechanicaltransduction in the mammalian left ventricle

Jacques LeF�evre *, Laurent Lef�evre, Bernadette Couteiro

LAIL, UPRESA CNRS 8021, Ecole Centrale de Lille, Cit�e Scienti®que, BP 48,

59651 Villeneuve d' Ascq, Cedex, France

Received 1 December 1998; accepted 1 August 1999

Abstract

We present a new lumped model of the pump behaviour of the mammalian left ventricle

based on simple but physiologically plausible sub-models of chemo-mechanical energy trans-

duction in muscle, mechano-hydraulic energy transduction in the ventricular wall and hemo-

dynamical coupling of the ventricle and its arterial load. The model builds upon the

foundation of classical analog ventricular models (dynamic compliance and visco-elastic mod-

els). However, we show that these classical models are not coherent from an energy viewpoint.

To insure this coherency, we introduce explicit cross-bridge mechanisms linked to the mec-

hano-hydraulical part of the model by a two-port capacitive (2PC) transducer representing

chemo-mechanical coupling. We show that this 2PC is thermodynamically plausible and, when

coupled to dissipative models of chemical energy generation and transfer, provides a novel and

consistent characterisation of cardiac energetics at the global pump level. Finally, we brie¯y

discuss some generalisations using nonlinear elements described by functional equations to

represent muscle memory and sur-activation.

It is a well-known fact that languages shape perception. Of all the lumped modelling lan-

guages, the bond graph (BG) method is the only one to use the notion of a 2PC as a primitive

modelling concept. Our hypothesis (mental model) is thus directly inspired by the fact that we

use the BG language. Our claim is thus that our work demonstrates very clearly the heuristic

and descriptive power of BGs in shaping new ideas about multi-energy and nonlinear physi-

ological applications. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Cardiac dynamics; Chemo-mechanical energy transduction; Pump function; Bond graphs

www.elsevier.nl/locate/simpra

Simulation Practice and Theory 7 (1999) 531±552

* Corresponding author. Fax: +33-3-20-33-54-18.

E-mail address: [email protected] (J. LeFeÁvre)

0928-4869/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 8 - 4 8 6 9 ( 9 9 ) 0 0 0 2 3 - 3

1. Introduction: current status of bond graphs in life sciences

As demonstrated by other papers in this special issue of SIMPRA, bond graphs(denoted frequently hereafter by the notation BG) are ideally suited to the modellingof nonlinear, multi-energy systems. Biophysical and physiological systems belong tothis category. We should thus expect a wide use of BGs in these ®elds but this isclearly not the case. After an initial surge of interest in the early 1970s [1], the useof BGs in life science modelling has become quite uncommon. In our opinion, thisis mainly due to two reasons:· In biology, an energy-based approach is often not at the right epistemological level

(i.e. it uses physical concepts ill-adapted to the level of understanding or experi-ments which we want to achieve). For instance in biochemistry, metabolism or en-docrinology, a BG model based on a mechanism-based or thermodynamics viewof the chemical reactions under study is often too detailed for our needs (e.g.studying tracer dynamics) or to cope with our experimental limitations (e.g. un-availability of heat or potential measurement). We usually satisfy ourselves witha phenomenological kinetic approach which does not use the concepts of energyor power and is thus inappropriate for BGs [2,3]. Specialists may use direct math-ematical formulations (e.g. kinetic di�erential equations) but biologists prefer amore graphical and intuitive method and use compartmental graphs or ForresterSystem Dynamics [3]. However, these graphical methods are not a panacea. Forlarge systems, they often result in very complex and unmanageable graphical rep-resentations with too many intermingled loops (the spaghetti syndrome).

· Even when physical, mechanistic and energy-based concepts are appropriate (e.g.hemodynamics, respiration, energy coupling in cells, photo-reception, transduc-tion, osmosis, cell excitability), life scientists feel uncomfortable with BGs. Theyconsider them as too abstract (as one of them put it quite bluntly to the authors. . . ``they are just a more mathematical Mickey Mouse''). They prefer thus electri-cal analogs [4] despite their awkwardness in representing multi-energy models.With respect to the ®rst problem, one of the authors (JL) has recently developed a

graphical modelling language using BG-like notations but in a non-energetic, purelykinetic framework [5]. This approach, called ``Transformation Kinetic Graphs'' re-sults in parsimonious and intuitive models and goes thus a long way towards solvingthe ®rst problem. However, it is not the goal of this paper to discuss it. We will in-stead focus on the second problem: overcoming the reluctance of biologists to the useof true BGs in what should be some of their natural applications. To convince them,what we need is the development of ``killer applications'', BG models showing thesuperiority of the method in biological and physiological multi-energy studies. Thesemodels should help us to make two points:· To show that, instead of being more abstract, BGs may in many cases provide

more intuitive solutions than other methods of graphical modelling. We shouldshow for instance that these models correspond more closely to our graphicalmental models of the interactions at work in the system under study than for in-stance analog networks or block diagrams. To make this point we should for in-stance show that the parts and interconnections of a BG model are more similar to

532 J. LeF�evre et al. / Simulation Practice and Theory 7 (1999) 531±552

the kind of ``back of the envelope'' drawings built by life scientists in their infor-mal discussions than analog or block diagram models.

· To show that BGs may lead to innovative modelling ideas di�cult to reach byusing other methods. Here it might be objected that BGs are simply a representa-tion of the underlying equations and thus cannot, epistemologically speaking, sug-gest new ideas. We claim the opposite. Languages shape perception andexpression. Some ideas are more easily expressed and/or suggested to the mindwhen using a language with the appropriate concepts and relationships. 1 We hopeto show that BGs have just the right high level and sophisticated primitives for en-ergy-based biological models and thus combining these concepts lead to innova-tive models more easily than with a less adapted language.In the following, we will present what we see as such a candidate ``killer'': a model

of the left ventricle showing the interaction of the determinants of ventricular pumpfunction (use of ATP, contractility, ventricular wall, hemodynamics). We will seethat, far from being abstract, the structure of the model is natural and translatesthe informal, back of the envelope diagrams drawn by cardiovascular physiologistsin their everyday work. We will show that this is due to the use of multi-energy trans-duction elements unique to BGs. This will thus make the ®rst point mentionedabove.

We will also show that the model suggests a new hypothesis on chemo-mechanicaltransduction in contractile proteins. This transduction will be seen as resulting fromthe operation of a device unique to BGs (what we have called a primitive of this lan-guage), a two-port capacitive transducer operating between the chemical and me-chanical domains. We will thus formulate for the ®rst time in the literature alumped cardiac model which is energetically coherent 2 and based on known mech-anisms but avoids most of the intricacies of detailed models. This provides in ouropinion an important step towards the solution of the current problems felt by peo-ple trying to integrate simple descriptions of mechanics and energetics at the muscleand ventricular levels. Indeed, our model provides fresh approaches on many ad-vanced features of global pump function: linearity of pressure±volume relationshipsversus nonlinearity of muscle force±length characteristics [6], simple energetics at theventricular level versus complex energetic slippage at the muscle level [7], role oflength-dependent deactivation at muscle and ventricular levels [8]).

A preliminary version of this model has been published in [9]. However the chem-ical part of this model was too simple to be implemented and no results were given.

1 A good but perhaps apocryphal example of this is that the Inuits do have 80 words for varieties of

snow, allowing thus a much richer and easy dialog on snow than in English although the same ``snowy''

ideas may no doubt be expressed in both languages.2 In the present paper, we will only discuss the conversion between chemical, mechanical and hydraulic

energy forms. We will thus neglect the thermal aspects despite their important impact on heart

performance. However, our representation of the chemical aspects of the model by reaction (i.e.

dissipative) two-ports provides the right structure to extend later the model in the thermal domain by

adding thermal ports to the reaction two-ports and using appropriate equations and models of the thermal

phenomena.

J. LeF�evre et al. / Simulation Practice and Theory 7 (1999) 531±552 533

The present version develops the chemical part and presents our implementation aswell as simulation results. In addition we discuss both the modelling and physiolog-ical relevance of our work quite in detail. Section 2 presents some notions about leftventricular function and discusses previous, non-BG-based models. The next twosections present our model. Section 5 brie¯y discusses model implementation andsimulation. Section 6 discusses how our model contributes to the general problemof convincing life scientists of the usefulness of BGs.

2. A primer on analog models in cardiac dynamics

Many experiments (see [10]) have suggested that, in normal conditions, the time-dependent values of the pressure and volume generated by the left ventricle (LV) areapproximately related by:

plv�t� � a�t�pa�Qlv� � �1ÿ a�t��pr�Qlv�; �1�where plv and Qlv are respectively the left ventricular pressure and volume: pr and pa

are nonlinear functions of Qlv describing the experimentally measured pressure±volume curves characterising the ventricle in completely relaxed (r) and active (a)conditions. Finally, a�t� is also an experimentally measured function. It is periodicbetween 0 and some level amax < 1 and describes the degree of activation of theventricle (a period � one heart beat) [6]. From a modelling viewpoint, Eq. (1) relatesa pressure and a volume and describes thus a time-varying or dynamic compliancecalled hereafter DC (note: the incoherent character of such an element from anenergy viewpoint is well known to bond graphers and will be discussed later). TheDC model represents two aspects of the LV behaviour (see a in inset, Fig. 1):· Contraction changes ventricular sti�ness from soft to sti� during each contraction

(increase of a) and generates the elastic potential energy used to propel the bloodduring ejection.

· During relaxation, the sti�ness goes back from sti� to soft to provide LV with thehigh compliance needed for its ®lling by the low pulmonary pressure during dias-tole.In a pressure±volume plane (Fig. 1), (1) describes a family of isochronal curves

(one for each value of a, i.e. for each time t in a contraction). Independently of its®lling and ejection conditions, LV is, at any time t, described by a point situatedon the isochron corresponding to t.

The full DC-model is shown in Fig. 2a. Two diodes represent the mitral and aorticvalves of the heart and are connected to the DC element denoted C�t�. The mitralvalve is connected to a model of the ®lling conditions (e.g. in the simplest case, animperfect pressure source). The aortic valve is connected to a model of the ejectingconditions (e.g. a simple network with resistive and capacitive features). During asteady periodic state, the operating point of the DC element describes thus a closedcurve in a pressure±volume plane. This curve is called a pressure±volume loop. Fig. 1shows two such loops corresponding to di�erent ventricular volumes and beat histo-ries but each ful®lling the isochronal conditions).


Fig. 1. An idealised view of left ventricular pumping in a pressure±volume plane. The abscissa is Qlv, the

ventricular volume and the ordinate is plv, the ventricular pressure. The interrupted lines are isochronal

curves given by Eq. (1) for speci®c values of the periodic activity variable a (see time dependency of ain the inset). The value a � 0 corresponds to the resting ventricle (curve pr); the value a � 1 corresponds

to the ventricle in maximal activity (curve pa). The intermediate isochrons are obtained for intermediate

values of a. During each beat, the sti�ness of the ventricle is initially given by the resting curve, then de-

velops progressively to reach the maximal activity curve (contraction) and then goes down to the resting

curve (relaxation) where it stays for a while before starting the next beat. At each time t during a beat, the

ventricle is described in this diagram by a point on the isochron corresponding to the value of a at that

time. Since steady state cardiac pumping is periodic, this operational point must describe a closed curve

during a beat. Such a closed curve, called a PV-loop is illustrated by the continuous line surrounding

the shaded area and traced in the direction of the arrows. The ventricle, ®lled by previous beats, starts

at the point r (relaxed) then contracts iso-volumetrically (vertically) until it opens the ejection valve. Then

the operation point goes to the left (ejection � decrease of volume). The pressure continues to increase,

reaches a maximum and then decreases until the ventricle reaches the state of maximal activity (point a) at

which ejection is maximal (volume is minimal), After that, the ventricle relaxes iso-volumetrically (pressure

goes down vertically) and reaches the relaxation curve where it ®lls for the next beat (the volume increases

on the resting curve) by accepting blood through its input valve to reach again the initial condition (point

r). The mechanical work done during a beat is the area of the PV-loop (shaded area) and the volume eject-

ed (stroke volume) is the di�erence between the maximal (point r) and minimal (point a) volumes. This

stroke volume multiplied by the heart rate (number of beats per minute) gives the ventricular output ¯ow.

Dependent on the ®lling and ejecting conditions (pressure and ¯ows imposed by the hydraulic networks at

the input of the mitral valve and at the output of the aortic valve), the ventricle may operate on many such

loops. It may for instance have high or low volume, large or small ejection, large or small work, large or

small pressure. However, in all these possible loops, the isochronal conditions imposed by Eq. (1) are al-

ways approximately veri®ed. For example, we give here a second degenerated loop observed at low volume

and in conditions in which the ventricle cannot eject (iso-volumetric � vertical). This is obtained for in-

stance if we clamp the aorta (quite a drastic manoeuvre). The ventricle starts from the point r1, increases

its pressure until a1 and then goes down to r1 again without any ejection. Despite the enormous di�erences

in behaviour, both loops verify the isochronal conditions. The points r1 and r are on the resting curve; a

and a1 are on the curve of maximal activity and the points reached t ms after the beginning of each beat

(i.e. c and c1) must be on the isochron corresponding to t.


Simple as it is, this description cannot be a complete model of a complex systemlike LV. Eq. (1) is just an approximation. Closer study reveals other e�ects with im-portant physiological and clinical consequences. For instance, (1) is satisfactory forsmall ¯ows but not for fast dynamics [8]. In this case, the minimal model (Fig. 2b)has four elements: a DC or time-varying compliance C�t�, an apparent resistanceR representing the experimentally observed relation between ventricular pressureand ¯ow, a parallel compliance CP and a modulation M of the active state a by ¯owand volume [8]. This model, called hereafter the generalised dynamic compliance(GDC) is purely phenomenological. Its structure results from a data ®tting exercise

Fig. 2. The simple and generalised dynamic compliance models. The upper panel (a) gives the simple dy-

namic compliance (DC) model described by Eq. (1). The left side shows the electrical analog and the right

side the corresponding BG. This DC model is made simply of a time-varying compliance C�t� placed be-

tween two diodes D1 and D2 simulating the mitral and aortic valves. C�t� is described by Eq. (1). The com-

pliance (analog to capacitance and thus inverse of sti�ness) is high when a � 0, decreases during

contraction, is minimal at full activity and re-increases during relaxation to stay low during the resting

phase. Blood (analog to electrical charge) ¯ows from the input (point A) into C�t� when C�t� is high

and is ejected from C�t� through the output valve in the aorta (point B) when C�t� decreases thus increas-

ing the pressure (analog to potential) across C�t�. When connected to appropriate models of the input and

output hydraulic conditions (called preload and afterload in physiological language), this model generates

PV-loops similar to those of Fig 1. These preload and afterload are indicated by the nodes called input and

output sub-models in the BG and by A and B in the analog network. The lower panel (b) describes the

generalised dynamic compliance model modifying the DC model to take into account fast ¯ow dynamics

(mainly force±velocity characteristics and deactivation). New elements have been added to the model of

panel a: R describing the relation between pressure and ¯ow observed experimentally; CP correcting

the resting elastic behaviour and M giving a modulation of a�t� as a function of the ventricular ¯ow

(F) observed experimentally and leading to a decrease of the maximal value of a when the ¯ow is large.

With appropriate values for its parameters, this model describes correctly ventricular dynamics in a larger

range of experimental conditions than the DC model. However, its structure is more complex and purely

empirical without any structural correlation or meaning in terms of known mechanisms.


and its various constitutive equations are complicated. It describes many experimen-tal results but the physiological meaning of its elements is unclear since its topologydoes not relate easily to well-speci®ed features of ventricular mechanisms and anat-omy:· The resistance lumps together two e�ects of ¯ow: a true (i.e. heat generating) but

small viscosity and an auto-regulatory mechanism intrinsic to the cardiac muscleand which does not generate any heat at all but decreases the force generatedwhen the contraction velocity increases. This second mechanism is thus seen inthe mechanical domain as a nonlinear viscosity (relation force±velocity) buthas no thermal counterpart. It is thus an apparent viscosity corresponding notto the viscous loss of a force already generated but to a velocity-sensitive lackof force generation. A simple element R cannot be related to these real mecha-nisms.

· Thermal aspects of chemical reactions are ignored and, as explained above, notrepresented by the power dissipated in R.

· The notion of C�t� is energetically meaningless since it does not take into accountthe energy needed to change the sti�ness and pre-supposes that this energy is in-dependent of the particular pressure and volume encountered in a beat (i.e. ofthe operating zone of the pressure±volume loop) and thus of ®lling and ejectionconditions. The use of a C�t� element is pragmatically justi®ed in models with asmall or irrelevant pumping energy (e.g. in electronic networks) but not in our casewhere it is the only source of energy.

· Finally, the modulation is given by a complex and empirical equation without anyinterpretation.Today's research on cardiac function puts emphasis on the relationships between

mechanics and energetics. Ideally, models should thus incorporate balances of all theenergy forms at work in the ventricle. In principle, complex models representing de-tailed cellular mechanisms, contractile proteins, excitability propagation and archi-tectural integration of the orientation of cardiac ®bres might address this need[10]. Some models along that line have been developed by various authors. They in-corporate appropriate relationships for chemical, mechanical and hydraulic energyand, usually neglect thermal aspects. At ®rst sight, this may seem surprising sinceheat has clearly a large share in total energy and a complex variation over a beat.Its elimination in the models is obviously due to the fact that cardiac heat generationis an elusive quantity, di�cult to measure and that many of its basic determinantsare imperfectly characterised. However, direct correlation has been found experi-mentally between oxygen consumption and mechanical work. This led people to hy-pothesise that in ®rst approximation, the step of chemo-mechanical transduction waslargely independent of heat which was mainly generated during the previous energy-generating steps made purely of chemical reactions [7]. They were thus able to ignoreheat by using a kinetic model for the chemical part and focusing then on a multi-en-ergy model of how the resulting chemical energy is transformed into pumping energywithout appreciable heat generation. In this paper, we will follow the same line al-though we will provide a natural entry point for a study of heat if and when the nec-essary data become available.


However, these complex models are unsuitable for our purpose of direct and easycomparison with `àverage values'' obtained from experimental or clinical studies. 3

They need too much data for their validation. We need to complement them with amodel simpler than the complex ones but more closely related to real phenomenathan the DC/GDC models. Such a model would serve as a useful bridge betweenthe understanding available in complex models and the limited data available inclinical or physiological studies of the ejecting heart. Since as explained before,our work, in its current stage, does not consider heat generation, our model shouldincorporate balances of all the other relevant forms of energy (mechanical, hydrau-lic, chemical). We will describe this model in Sections 3 and 4 will discuss its equa-tions, emphasising its main innovative feature, a chemo-mechanical capacitivetransducer.

3. A simple BG model of the isolated ventricle

3.1. Informal description of the mechanisms incorporated in the model

Fig. 3 illustrates a physiological preparation in which a canine left ventricle is iso-lated from the body and attached to glass pipes in a glass box B providing adequateexternal conditions and coronary metabolic support from a support dog [6,8]. Theventricle is represented by a variable compliance (LVC) which, due to the musclein its wall, alternates between soft and sti� values. This LVC is attached to glasstubes connected to its valves (MIV � mitral � input and AOV � aortic � out-put). MIV is connected to an input tank (IT) ®lled with blood and providing the di-astolic ®lling pressure. AOV is connected to a rubber bag simulating aorticcompliance (AC) and to a narrow glass tube simulating peripheral resistance (smallvessels PR). The pressure responsible for blood ¯ow is created by the wall of theLVC (i.e. LVW) in which muscle ®bres contract to generate a mechanical powertransduced by LVW into hydraulic power available in LVC. The tubes connectingIT, LVC, AC and PR are supposed to be ideal.

In the real ventricle, millions of ®bres are arranged in a complex pattern similar toa rope winding around the cavity. These ®bres contract at slightly di�erent times butin a well-determined sequence. In our model, we represent all these ®bres by a singleequivalent ®bre (EF) composed of two protein ®laments (myosin, actin). The thickmyosin ®lament has millions of small extensions (cross-bridges X) pointing towardsthe actin. At rest, the two ®laments are detached or free (index d or f). During con-traction, the detached cross-bridges use the energy provided by a supply of ATP to

3 Let us remark that we emphasise the word `àverage''. Our model will contain too many parameters for

identi®cation from data measured in a single preparation or patient. These parameters will thus have to be

collected from various sources and will represent typical normal or pathologic situations. The use of our

model will thus be as an `ùnderstanding help'' in comparative studies with the average behaviour of a class

of preparations, experiments or patients and not in characterisation or control of individual experiments

or patients.


attach to the actin ®lament (index a). They become detached again during detach-ment. Each cross-bridge attaches and detaches several times during a beat. Globally,this results in a periodic variation of the total number of cross-bridges attached at agiven time. This number is low in relaxation (diastole) and high during activity (sys-tole). Its period is equal to the duration of the cardiac beat. When it is attached, abridge is able to act as a small stretched spring having some elastic potential energyand thus able to generate force and shortening. The global process of attachmentand detachment is represented by a chemical reaction between Xd and Xa, the globaltime-dependent numbers of detached and attached cross-bridges in each state. Thisreaction has periodically varying kinetic constants ka and kd dependent on muscle ve-locity (see the thin arrow � velocity modulation of the reaction rates). This resultsin a periodically varying value of Xa and thus in a periodic variation of the global

Fig. 3. A ``back ot the envelope'' view of the multi-energy left ventricular system. The ®gure shows a prep-

aration in which the ventricle is isolated from the body and connected to arti®cial tubing to provide ad-

equate input and output conditions. The left ventricle is represented as a compliant bag (LVC) between its

two valves (MIV, AOV). LVC is ®lled from an input tank (IT) at constant preasure and ejects into an hy-

draulic load made of an elastic bag simulating the aortic compliance (AC) and a narrow tube simulating

the resistive obstacle constitued by the small peripheral vessels (PR). The LVC is in a box B at appropriate

temperature and ionic conditions and receives coronary support (blood ¯ow) from a donor dog. The wall

of LVC (LVW) is elastic with a variable sti�ness. It is made of muscle energised by chemical reactions de-

scribed in the main text and represented in the actin±myosin box. This box shows the reactions of attach-

ment and detachment of cross-bridges between the actin and myosin ®laments of the cardiac muscle. These

reactions are periodic (periodically varying rates ka and kd) and may be dependent on the muscle velocity

(interrupted arrow). ATP is needed to drive the attachment reaction. The attachment of the bridges results

in the creation of small stretched springs in the muscle increasing thus the sti�ness of the muscle and pro-

viding the basis for the development of the potential energy for contraction. This is indicated by the con-

tinuous arrow between the actin±myosin box and LVW. Classically this arrow is seen as unidirectional

from acto-myosin to the wall. In our paper, we argue that it might be necessary to consider the reverse

coupling and thus we make the hypothesis (question mark ?) that this arrow is bidirectional.


number of little springs in LVW, i.e. of the sti�ness and available potential energy ofthe equivalent muscle.

This chemo-mechanical coupling is represented in Fig. 3 by the bidirectional arrowlinking Xa to the elastic LV wall. Classically, this coupling is considered as unidirec-tional, a modi®cation of Xa modifying sti�ness and thus muscle length and ventricularvolume. However, in the following sections, we will introduce the hypothesis of a bi-directional coupling and show that its formulation and simulated behaviour are co-herent with experimental data. In our formulation, a change in muscle length willcreate a reverse change in the chemical backward a�nity of Xa and thus a�ect the at-tachment±detachment cycle. Compatibility with data on pressure and ¯ow is certain-ly not su�cient to validate our hypothesis which needs further experimental andtheoretical study. However, we would like to remark that the changes of chemical po-tential hypothesised above should in¯uence the cross-bridge cycle and thus ATP con-sumption. It could have an impact on e�ects like the ATP non-Stoichiometricslippage observed by Gibbs [7] or an e�ect similar to the slight reverse ATP synthe-sis demonstrated in skeletal muscle by Gillis and Mar�echal (see [7] for references tothis work).

3.2. The BG model of the complete system

Fig. 4 shows the BG corresponding to the system of Fig. 3. The model is organ-ised in four connected panels representing di�erent energy domains or transductionmechanisms.

Hydraulic or hemodynamics sub-model: The hydraulic sub-model (panel a) is verysimple: SE is a constant pressure source representing the input tank. The elementsRd are ideal diodes representing the valves (MIV � mitral and AOV � Aortic).R and C are linear resistive and capacitive elements representing the hydraulic loadmade of the rubber bag and the peripheral resistance. Connecting pipes are seen asideal connections and inertia is neglected. The 0-junction connected to the lowerpanel represents the connection to this system of the ventricular chamber representedby these lower panels. All together, this sub-system is quite uninteresting. If we con-nect a time-varying C to the 0-junction to replace the lower panels, we obtain theclassical DC model (see Fig. 2a). If we limit ourselves to this hydraulic level, theuse of bond graphs is nearly equivalent to the use of electrical analogs (see alsoFig. 2). The BG model nevertheless presents a slightly more intuitive topology sinceit does not need to represent explicitly a line of pressure reference. In an electric an-alog model, this ground line needs to be represented but does not transmit any ¯owsince all its points are at the same pressure. It is thus useless and creates confusion inthe mind of the user of the model since it has the same modelling status than a real¯ow line but without any hydraulic ¯ow. This is not important when the model issimple but becomes more signi®cant if we need to represent the whole cardiovascularloop (not illustrated here) with several reference pressures (extra-thoracic, thoracic,pericardial and even septal). Despite this slight representational advantage, the con-sideration of the hydraulics domain alone is certainly insu�cient to convince life sci-


Fig. 4. The complete multi-energy BG model corresponding to Fig 3. The four sub-panels correspond to

clearly separated mechanisms or energy domains. The lower panel (d) shows the degradation of the ener-

gy-rich molecule ATP into ADP. The energy freed during this conversion is used to energise the reaction

occurring in the next panel (c). In this panel, we see the simplest possible model of the complex chemical

process leading to the time-varying attachment and detachment of actin A and myosin M in acto-myosin

AM. The model used is an elementary reaction R with time-varying kinetic constants. As indicated in the

text, heat generation is not considered since we focus mainly on the energy balance during the later step of

transduction (see panel b). However, a thermal bond is attached to R to show where heat will be ex-

changed in further models considering the thermal aspects of R. It must be remarked that both ATP-

ase and contractile proteins have been included in a single reaction mechanism. The element C on the right

side of the reaction is a 2PC element with port 1� bound AM in the chemical domain and port 2 � x in

the mechanical domain. Port 2 sends the elastic potential energy of AM stored in this two-port C to the

next panel (b). This panel b represents the mechano-hydraulic energy conversion occurring in the ventric-

ular wall by a simple power-conserving transducer TD which receives mechanical energy on its port 2 and

send the transduced hydraulic energy through its port 1 in the upper panel. Stray elements (interrupted

lines) may be added to represent small viscous and inertial aspects in the mechano-hydraulic conversion.

The upper panel (a) represents the hemodynamic mechanisms: a pressure source SE develops the pressure

needed to ®ll the heart (i.e. the hydraulic port of TD) during diastole. A parallel connection of a compli-

ance C and a resistance R represents the arterial load. Both SE and the load are connected to the ventricle

through valves represented by ideal diodes Rd (mitral�MIV and Aortic�AOV). As explained in the

main text, the topology of this model is very similar to what ``wet'' physiologists see in their mind when

thinking about this system. This is in our opinion the ®rst advantage of BGs. The second advantage is

clearly its multi-energy nature which allows us to use elements like TD and AM.


entists of the ``unique'' advantages of BGs. Things become di�erent when we startlooking at the lower panels representing energy transduction.

Mechano-hydraulic transduction: The ®rst step of this transduction (panel b) is torepresent how the LV hydraulic energy needed in the hydraulic sub-model is ob-tained from the mechanical energy generated by an equivalent muscle representingall the muscle ®bres. This operation is done by the LV wall and, neglecting secondorder e�ects, we represent it by a power-conserving transducer (panel B). Indeed, in-ertial e�ects are certainly present but small. Moreover, viscous losses due to frictiondo certainly occur in the wall but they are small compared to the energy transduced.In ®rst approximation, we may thus equate the mechanical and hydraulic energy. Ifour simulation experiments show that this is not su�cient to correctly represent thecardiac pressure and volume wave shapes, we will later add parasitic inertial and vis-cous elements to this TD (see elements Iw and Rw connected by interrupted bondsand neglected in the model used in this paper). Under these approximations, the wallis thus a pure TD i.e. a device transducing pressure and ¯ow at constant power. Toget its constitutive equations, let us suppose that we can determine empirically an al-gebraic function uG (G for geometry), dependent on the geometry and kinematics ofthe ventricle and relating the volume Qlv and the length lm of the equivalent muscle atany time t in a cardiac cycle:

Qlv�t� � uG�lm�t�� $orlm�t� � uÿ1

G �Qlv�t��: �2�This is easily found for simple shapes and kinematics (e.g. contracting cylinders orspheres) but more di�cult if realism is needed. In that case, it might be preferable touse an abstract formula ®tted from real data. Eq. (1) gives then a relation betweenLV ¯ow and muscle velocity:

dQlv

dt� wG�lm� dlm

dt; �3�

where wG � duG=dlm. As explained above, we suppose, for simplicity reasons, thatthe transformation between force and pressure is lossless and has a negligible inertia.Equating then the mechanical and hydraulic powers and using (3), we get a relationbetween ventricular pressure plv and muscle force fm:

plv � �1=wG�lm��fm: �4�Eqs. (3) and (4) describe a non-energic, i.e. power-conserving transducer (TD); port 1is in the hydraulic domain and port 2 in the mechanical domain. This element is a ®rstexample of the way in which BGs provide just the right primitives to express the kindof models we need. Indeed, this TD is a slight extension of a classical BG element. Itsuggests thus itself quite naturally to every bond grapher. However, it has noequivalent primitive in other modelling formalisms and, as illustrated by the fact thatthey have not been previously described, people using these formalisms will thus hitwith more di�culty on the general equations given in Eqs. (3) and (4). The fact thatwe need such an element, natural in BGs and exotic (but not impossible to conceive)in other methods is a ®rst indication of the usefulness of BGs in our problem.


The true power of BGs will still become more obvious when the chemical andmechano-chemical processes will be modelled in panel C. Again this will need the in-troduction of typical BG notions which do not come naturally to the mind whenusing other methods.

Chemo-mechanical transduction: Our next step (panel c, right side) is to representin the simplest possible way how the muscle makes the transduction of the chemicalpower of attached cross-bridges into mechanical power (panel C). As a ®rst approx-imation, we will consider this transduction as resulting from the action of a two-portnode called AMCX in panel c. As explained before, we will neglect here the thermalaspects which, according to our hypothesis explained above, belong not to transduc-tion from Xa to mechanical power but to chemical reactions from Xf to Xa (represent-ed here by a kinetic process) and to all other calcium-related reactions and cellularmaintenance processes.

The port X (attached cross-bridges acting as springs) of this two-port element isconnected to the mechano-hydraulic transducer TD described above to which it de-livers, after transduction, the energy received from its other port (AM � acto-my-osin) and coming from the chemical reaction of the cross-bridges. The node AMCX isthus a two-port transducing chemical energy into mechanical energy. 4 This two portmay be simple or have an internal structure made of many elements. We will makethe simplest hypothesis and consider it as an elementary two port. Seen from the ven-tricular side (index 2), it must provide elastic potential energy and thus behave as avariable spring X, i.e. a C-element (variables: lm, dlm=dt, fm). On the other side (in-dex 1), it must receive and store a concentration of attached and thus energy-richacto-myosin cross-bridges from the model of the cross-bridge attachment±detach-ment reaction given in the left side of this panel c. In BGs, a store of chemicals isalso a C-element (variables: Xa � attached cross-bridges, dXa=dt � reaction speedand la � potential). The simplest elementary muscle transducer is thus a two-portcapacitive transducer. We call it a 2PC. Seen from the mechanical side, it should notbe very di�erent from a time varying capacitor since we know that the DC model is areasonable ®rst approximation. Calling Xa the number of attached cross-bridges, wecould for instance adopt an element described at the mechanical port by an equationlike:

fm�t� � a�Xa�t��fa�lm� � �1ÿ a�Xa�t��fr�lm�; �5�where fm is the muscle length, a�Xa�t�� is a pumping variable varying from 0 (re-laxation) to 1 (attachment) with the periodic variation of Xa, fa and fr are the twofunctions of muscle length �lm� giving respectively the active and resting force±lengthcurves.

Indeed, using (4) and (5), (2) gives then a relation between Qlv, plv and t which isvery similar to the previous DC model. The only di�erence being that Xa is not animposed function of time but depends on the chemical model. Obviously, we will also

4 The notation AMCX is our only departure from standard BG notation. We use it to emphasise the non-

classical nature of the two domains between which this C operates (chemical and mechanical energy).


need an equation for the chemical port and we will see in Section 4 that its choice isfar from trivial.

As explained above, this is only the simplest hypothesis which we can make on thenature of the chemo-mechanical transduction step. If comparison with experimentalresults indicate that it is oversimpli®ed, we may add stray elements to represent forinstance viscous losses or a multi-step transduction.

At this stage, it is perhaps worthwhile to remark once more that a 2PC-element,linking two di�erent energy domains, may only be easily conceptualised if we useBGs. Again, the introduction of this element in our model is thus a direct conse-quence of the BG formalism as evidenced by the fact that such an element, althoughquite natural has never been considered by other modellers.

The chemical cross-bridge cycle: Like before, we will use the simplest chemicalmodel able to produce a periodic attachment and detachment of the cross-bridgesX (see in Fig. 4, panel c, left side, a classical model of a chemical reactionAM free $ AM bound. The chemical equation of this model will thus be:

Xd

!ka

kd

Xa; d � detached; a � attached �6�

in which ka and kd are periodically varying kinetic constants of attachment anddetachment. As a ®rst and heuristic approximation, the potentials are given bymodi®ed Nernst formulas:

la � �Aa � Ba�T � ln�Xa��b and lD � AD� � BD�T � ln�XD��: �7�

These equations need some justi®cation. They are valid in idealised conditions (di-luted solutions, ideal mixing, large volumes, non-compartmentalisation) which arecertainly unveri®ed here. We use them only as a ®rst approximation and our guess isjust that they should be su�ciently accurate for our purpose.

The only non-classical feature of (7) is the factor b (see below). Using the usualexpression of a reaction rate in chemical thermodynamics, the reaction ¯ow maybe expressed as:

dXa

dt� ka e�ldÿAd�=Bd ÿ kd e�laÿbAa�=Ba �8�

which by (7) gives a modi®ed mass-action formula:

dXa

dt� kaXd ÿ kdX b

a : �9�

b is thus a reaction order taking into account the fact that detachment is a multi-stepprocess. (Note: we have supposed here that attachment being much faster than de-tachment may be seen as a single step process.) This point might need modi®cationand a second reaction order could be needed for the attachment term. The completechemical model corresponding to Eq. (6) is represented by the chemical box of panelC (left side).


Some characteristics of this model are worth discussing.· Firstly, the two-port R represents the reaction mechanism and is the only element

introducing both an apparent mechanical viscosity and a true heat generation inour model.Apparent viscosity will result if, as it seems to be the case in reality, ka and kd arefunctions not only of time but also of muscle velocity, introducing thus a kinetic(i.e. non-heat generating) velocity dependence of the force generated in AMCX.True heat generation, although not described here will result if we add the neces-sary entropy or heat production equation to the constitutive equations of R. Wemay then exchange this heat with the outside world by adding a thermal port, il-lustrated but not used in panel c. It must be remarked that, for simplicity, we haveadded the ATPase e�ect of acto-myosin to the same R. In more sophisticatedmodels, this might be separated in a speci®c model. However, in our case, we onlytake into account the stoichiometry of this ATPase and not its energetics aspects.The full reaction is thus AM free� nATP! AM attached� nADP.Capacitors might as well be attached to the 1-junctions of their respective sideof the reaction for ATP and ADP.

· Secondly both Xa and Xd are represented by C-elements. As seen above, Xa is notan independent element but one of the ports of the 2PC AMCX. To complete thedescription of our model, we need now to come back to the 2PC.

4. The 2PC model of chemo-mechanical transduction

To the best of our knowledge, it is the ®rst time that a chemo-mechanical 2PC isproposed as a modelling element not only in physiology but also in the entire BGliterature. Its de®nition, its conditions of validity and the derivation of its constitu-tive equations will therefore be completely detailed in this section. We will proceedby analogy with a similar transducer, a moving plate electrical capacitor whichhas a well-known elementary theory [11]. However, the reader should realise thatour chemo-mechanical 2PC is just an hypothesis and a rather non-classical one.The following derivation must thus be considered only as tentative.

We will start by supposing a simple energy equation from which we will obtain theconstitutive equation by partial derivation. However, in our case, in contrast with themoving plate capacitor, the energy equation is not known and certainly more complexthan supposed here (if it is at all valid to consider a unit volume of highly inhomoge-neous muscle as having a well-de®ned characteristic energy function). We will adoptthe simplest possible form allowing a reasonable behaviour at both the mechanicaland chemical sides. Therefore, the following computations represent just a ®rst ordertheory and should be made more complex if and when warranted by simulation results.

To de®ne our chemo-mechanical transducer in analogy with the electro-mechan-ical capacitor, we must start from the energy equation of the 2PC. If it exists, it ismost probably a very complex functional (not even a function but a law dependenton the past history of the muscle). For instance, it might depend on anisotropy oflength change (see in [6] the discussion of Yin's work on cardiac muscle anisotropy),


on past history (see in [13] the discussion of Hunter's work on cardiac sur-activationand memory) and on many other features (e.g. velocity, other chemicals like Xf orintermediate states of acto-myosin). Since we want to start with simple relationships,we will neglect the energy in free cross-bridges and suppose that the energy of themuscle is simply a function of its length and of the number of attached cross-bridges.

With the above strong hypotheses, we may therefore suppose the following sim-pli®ed but general form:

Em � Em�lm;Xa�: �10�This is the basic equation of our element and we should not be deceived by itssimplicity. First, it is most certainly an approximation (see the discussion onfunctionals above). Secondly, even under our simpli®cation hypotheses, the com-plexity of the muscle structure and mechanisms suggests that the unknown functionEm cannot be simple. However, we will now obtain a reasonable ®rst approxima-tion by using data on the mechanical behaviour and simple thermodynamic hy-potheses.

From (10), we see by derivation that the power may be given by:

Pm � oEm

olm

dlm

dt� oEm

oXa

dXa

dt� fmmm � la

_Xa: �11�

The Maxwell equation (i.e. the equation expressing the reversibility condition due tothe exchangeability theorem on second order derivatives in a continuous multi-variable function) is:

ofm

oXa

��lm

� ola

olm

��Xa

: �12�

Then, by identi®cation of (11) and (12), we see that the force and the potential of Xa,i.e. the two constitutive equations of the 2PC are given by:

fm � oEm

olm

and la �oEm

oXa

: �13�

The two equations given in (13) are especially interesting from a physiological pointof view. The one on the left is classical since everybody admits that a change in thenumber of cross-bridges creates a variation of force at constant length. This is indeedthe whole idea behind every model of muscle contraction. However the equation onthe right is completely new. It should even be a bit of a shock to many physiologists.It shows that if our 2PC hypothesis is correct, there is an equal variation of Xa withlm. This could lead to a reverse ATPase and be a critical test of our model (see [7] forexperimental evidence).

To specify the constitutive equations (13), it remains only to choose the functionEm. Let us start by describing the experimentally determined force±length relationsof a muscle strip by (see (5)):


fm � EX �Xa�E0act�lm� � �1ÿ EX �Xa��E0pas�lm�: �14�EX is the active state function replacing the former a�t�. It varies between 0 and 1 as afunction not of t but of Xa (development of the number of springs) which variesperiodically due to the periodic character of the attachment and detachment rates.E0act and E0pas are the experimentally measured active and passive force±length curvespreviously called pa and pr. (Note: the explanation of our new notations for thesecurves will come below when we see that these functions are derivatives of thefunctions determining the length dependency of the energy.) Using (14), integrationof (13) gives:

Em � EX �Xa�Eact�lm� � �1ÿ EX �Xa�Epas�lm�; �15�

E0act�lm� � dEact�lm�dlm

and E0pas�lm� � dEpas�lm�dlm

: �16�

To choose EX �Xa�, we use Nernst law (7), (13) and (17). Again by integration we get:

la �oEm

oXa

� Eact�lm�ÿ ÿ Epas�lm�

�E0X �Xa�; �17�

b � Eact�lm�ÿ ÿ Epas�lm�

�; �18�

E0X �Xa� � Aa � Ba�T � ln�Xa�; �19�

EX �Xa� � AaXa � Ba�T �Xa ln�Xa�� ÿ 1�: �20�

The reaction order b is thus length-dependent and expresses the reverse Maxwellcoupling (exchange of second derivatives, see (12)) between the mechanical andchemical parts dependent on (16). Validity limits must obviously be introduced forXa to verify the monotonicity, positiveness and boundedness conditions of EX . Asexplained above, the preceding equations should be made more complex if necessary.Considering the mechanisms and phenomena usually represented in more complexmodels, it seems plausible that the following points should be introduced at furtherstages to make our model more realistic:· The functional form of the energy equation should include a thermal variable, a

dependency on detached cross-bridges and eventually a third cross-bridge state be-tween attached and detached.

· Eq. (14) and all its consequences should thus be accordingly modi®ed.· An order parameter should be introduced not only for the detachment but also for

attachment and eventually for the intermediate steps needed by more complexmodels.

· Velocity- and length-dependent parameters should be introduced in the cross-bridge cycle.


This section has introduced the main innovation of our model, the 2PC elementused to represent reversible transduction. For reasons of space limitation, we couldnot explain its derivation at full length. More complete explanations are available inan internal report available from the author. Otherwise, from what is given here, itshould be clear that this BG-inspired analogy between a moving plate capacitor anda chemo-mechanical process might, if validated, provide a fruitful new direction inlumped cardiac modelling. It is also clear that we could not have introduced this el-ement without knowledge of bond graphs since it is framed after the classical twoport C-element of BG.

5. Implementation and preliminary results

We have implemented our model using 20-SIM (a trademark of CONTROL-LAB, The Netherlands). Since most of our BG elements are not classical, sub-modelshad to be de®ned for some elements using SIDOPS, the object-oriented language un-derlying the graphical de®nition of BGs in 20-SIM. The full model is given in Fig. 5.At ®rst sight, it is complex and a far cry from an intuitive description close to themental models of the physiologists. However, by isolating mentally the BG part fromthe block diagrams, the reader will see the emergence of the structure of Fig. 4 whichin fact is very similar to informal diagrams used by physiologists to explain energytransduction and energetics of the heart (see [6,7]). By de®ning more modular sub-models we could have obtained a model much closer to the one given in Fig. 4. Thiswould be needed to make communication with experimental physiologists easier buthere it would just hide some features of the implementation.

We are in the process of evaluating our results in depth however, the ®rst simu-lations give clear signs of the plausibility and validity of the model. Our preliminarysimulations were only aimed at ®nding parameter values giving a qualitatively ac-ceptable behaviour. Nevertheless, typical results show pressure±volume tracingscomparable to reality (Fig. 6). Moreover our model shows a length-dependent deac-tivation e�ect similar to the one observed in [8]. It is a direct consequence of the or-der parameter and of the Maxwell reciprocity conditions. However, systolicdeactivation is still incomplete and needs further study. In order to completely repro-duce deactivation e�ects like those observed in [8], our current hypothesis is that wewill need some of the modi®cations mentioned at the end of the last section (mainlythe order on detachment) [13]. However, in its current state, the model describedhere provides already a better representation of PV-loops than classical models.

Recently, we have proposed new BG elements characterised not by algebraic re-lations between their basic variables but by functionals (i.e. applications from func-tions to numbers) representing fading-memory properties [12]. We have alsopresented a way to incorporate such elements in BG simulations while keeping ener-gy balances correct. One of our prime motivations was to represent the hereditaryproperties of cardiac muscle. Indeed, in addition to deactivation, modern experimen-tal studies show an opposite mechanism called sur-activation and dependent on pre-vious beats. According to recent work, the level of activation of a ventricle and thus


its capability of generating potential energy could be determined by a delicate bal-ance between these two mechanisms of deactivation and sur-activation. The intro-duction of this memory e�ect needs more complex models than the purelydi�erential ones described above. Currently, we test a model in which, in additionto the above mechanisms, we introduce kinetic constants dependent on a convolu-tion of the past muscle length on some time horizon with a fading-memory kernel[14]. Taking the above model as a reference, our ®rst results show an end-systoliccurve with higher pressure at a low volumes and lower pressure at large volumes.This could explain the aforementioned balance between sur- and deactivation.

Fig. 5. 20-SIM implementation of the model of Fig 4. Interrupted boxes indicate the hemodynamics and

chemical parts of the model. The ®gure seems complex but, by making abstraction of the block diagrams,

the reader will see the basic BG structure of Fig. 4. Some BG elements speci®c to our model have been

de®ned directly by text code (20-SIM sub-models). They are indicated by ovals: 2rl is the resistive two port

computing the acto-myosin attachment±detachment rate; cfp1 is a ¯ow source used in some simulations to

represent a syringe injecting or withdrawing a ¯ow pulse directly from the heart. Remark the element 2cl

which is our capacitive two-port. The block diagram elements compute all the auxiliary relations of the

model described in the main text. The block energie3 computes in addition the energy and power generated

in a beat.


6. Conclusion

This paper has introduced a model of the left ventricle which implements what isessentially a new ``macro-interpretation'' of the ventricular capacity for generatingelastic potential energy. Instead of being unidirectional like in all previous models,the chemo-mechanical coupling is seen as reversible, storing potential energy andclosely linked to the contractile chemical events. However the generation of chemicalenergy itself remains obviously irreversible and heat generating. Although heat is notconsidered here, the model shows the proper structure to allow its study in a furtherphase.

Much validation work remains to be done. However, the preliminary results arepromising and at least compatible with real data. The DC concept, explaining qual-itatively so many aspects of LV dynamics, remains central to our analysis but it isgeneralised in an energetically coherent way. If our model survives its tests, it will,

Fig. 6. Typical pressure±volume loops obtained in the control state. The Figure shows ®rst the maximal

activity line (AB) obtained in a non-illustrated sequence of iso-volumetric beats obtained by starting beats

at di�erent initial volumes and by increasing very strongly the forward resistance of the aortic valve. We

also show the passive sti�ness curve obtained in these conditions. Finally, we show a series of ejecting PV-

loops obtained in various ®lling and ejecting conditions (see the legend of Fig. 1 for explanations). These

loops behave approximately like in the DC-model. However, like in real beats, they do not reach the pas-

sive and active iso-volumetric sti�ness curves but stay inside the framework de®ned by these curves.


for the ®rst time provide a concise and rigorous framework for the study of LV en-ergetics at the global pump level. Our hope is that it will then be possible to link themicro-level descriptions available from computational models and the macro-levelbehaviour of the whole cardio-vascular loop.

In addition to its physiological interest, our model shows, in what we claim to be avery convincing manner, the unique power of BGs for biophysical and physiologicalmodelling. Its structure, close to those used by physiologists in their informal discus-sions, is indeed intuitive. We have also suggested that the use of BGs leads naturallyto new hypotheses which can then be tested by simulation and suggest critical exper-iments. This is due to the fact that, like in every language, the primitives and con-structions available in BGs shape our perception of the systems under study. Byemphasising energy-coherent models and by giving new primitive modelling bricksfor energy transduction, BGs provide just the right tools to express the kind of phe-nomena represented here. The same model can undoubtedly be built with othermethods (block diagrams, code. . .) but its basic notions would be less easy to comeby.

Finally, some of our elements are very peculiar (periodic reactions, chemo-me-chanical two-port, nonlinear transformer). The ease with which we implementedthem in 20-SIM is a very positive test of the maturity and openness of the currentblend of BG modelling packages based on object-oriented based languages (e.g. SID-OPS, MODELICA). It is our experience that using such packages leads to model de-velopment times at least an order of magnitude faster than with block-oriented orcode-based methods.

As claimed before, the development of BGs in biophysics and physiology needssuch models. We hope that the preceding study will serve as an inspiration for othersimilarly inclined researchers.

Acknowledgements

This work was done while the ®rst author was based at the Sherrington School ofPhysiology (now part of King's College Medical School), St. Thomas's Campus,University of London.

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a bond graph model of chemo-mechanical transduction in the mammalian left ventricle

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