Myocardial Contractions and the Ventricular Pressure–Volume Relationship

Paola Nardinocchi ∗ Luciano Teresi † Valerio Varano ‡ May 28, 2010

Abstract resented in the pressure–volume loop which, in the end, represents an assessment of the overall mechan- We present a reduced–order model with the ical activity of the heart; the reader is referred to the aim of introducing a novel point of view in the inter- excellent review (1), or to (2). pretation of the pressure–volume loops. The novelty Ventricular chamber pressures and volumes are re- of the approach is based on the definition of active lated through the contractile behaviour of the mus- contraction as opposed to that of active stress. The cular structure of the cardiac tissue. A qualitative consequences of the assumption are discussed with and quantitative bridge between the function of the reference to a specific pressure-volume loop charac- heart as a whole and the microscopic dynamics char- teristic of a normal human patient. acterizing muscle contraction would be desirable but it is still unrealized (3). There exists complex mathe- Keywords: Active deformation, non–linear elastic- matical models relating ventricular chamber pressure ity, . and volume to muscle length and force generation which are the basic notions characterizing the muscle PACS 46.05 +b, 62.20.D subclass MSC 92C10. function at a macroscopic level. Nevertheless, it is well known that reduced–order heart models would be helpful in a qualitative assessment of the global 1 Introduction state of health of the left chamber (4). Here, we present and discuss a novel point of view The heart is a specialised muscle that contracts reg- in the interpretation of the pressure–volume loops ularly and pumps blood to the body and the lungs. based on a special modeling of muscle tissues and The centre of the pumping function are the ventri- aimed to introduce new insights into the mechanisms cles. Due to the higher pressures involved, the left determining the performance of the ventricular cham- is especially studied. The pumping action ber. is caused by a flow of activation potential through The first issue is the modeling of the contrac- the heart that repeats itself in a cycle. The effective- tile behaviour of the muscular structure of the car- ness of the pumping action may be evaluated through diac tissue. From the microscopic point of view, the analysis of different parameters: the stroke vol- the excitation-contraction coupling (ECC) in car- ume, the , the end–systolic pressure– diac muscular fibres is a complex mechanism in- volume relationship, etc.... All of these are well rep- volving many variables such as membrane potential, ionic conductance, intracellular calcium concentra- ∗Universit`a di Roma “La Sapienza”, Roma, Italy, [email protected] tion, membrane strain and stress, and changes in the †Universit`aRoma Tre, Rome, Italy, [email protected] rest length of muscle fibres due to the interaction of ‡Universit`aRoma Tre, Rome, Italy, [email protected] actin and myosin. The definition of a realistic model

1 that is sufficiently complete to account for the prin- heart. Our modeling, with its special notion of con- cipal mechanism involved in ECC, yet simple enough traction, defines a correspondence between each point to be effective, is an arduous task. The issue has of a PV loop and the tissue contraction which de- been addressed from the microscopic point of view termined it. In the end, a PV loop may be viewed in various papers (5), (6), (7). However, the mus- as determined by three main state variables, pres- cle fibres ability to contract and relax (in response to sure, volume, and contraction; changes in ventricular biochemical signals) involves the function of the heart function visualized through PV loops may be easily as a whole. Hence, it is mandatory to have a macro- interpreted in terms of changes in these variables. scopic model of the contractile properties of cardiac Severe geometrical models of left ventricle are used tissues in order to discuss the pumping function of to estimate the overall mechanical activity of the the heart in terms of contraction as well as pressure heart. We refer to a spherical approximation of the and volume. left ventricle: the key mechanical relationships are Our modeling of muscle tissues accounts for muscle not markedly altered by the simplified geometry (1) contraction through the notion of active deformation. and we can easily present and discuss our different We assume that the contraction experienced by a fi- point of view (see (12) for more complex conceptual bre of muscle tissue under stimulus is described at the geometries). Moreover, we do not account for any macroscopic scale by a (stress–free) change, called ac- tissue anisotropy: both the passive and the active re- tive deformation, in the length of the fibre. We say sponse of the tissue are assumed to be completely that a muscle fibre stays in its active state when it is isotropic. Nevertheless, the large deformations of contracted; otherwise, it stays slack. It is worth not- the ventricle make it mandatory to set the modeling ing that, from a mechanical point of view, the active within the context of non–linear elasticity. state of a muscle fibre is a ground state as determined To better illustrate and discuss our point of view, by a change of length due to electrochemical stimuli. we use a specific sampling of pressure–volume pairs The visible state of the muscle fibre is determined measured by (15) with reference to a normal human from the active state by the amount of stress the fi- patient. We use the corresponding PV loop, shown bre sustains. Roughly speaking, the active deforma- in figure 6, to discuss the main characteristics of our tion describes how muscle tissues shorten once acti- reduced–order heart model. vated and are left free to contract, while the visible Let us note that here we are not interested in re- deformation describes the state that a muscle tissue lating heart electrophysiology to myocardial contrac- attains once contracted, loaded and/or kinematically tion; a proposal about this issue, based on the notion constrained (as in isometric activation). The mate- of active contraction, as been presented in (13), (14). rial model based on our modeling of muscle tissue is called the two-layer model (8), (9), (10), (11). The visible layer, which is observable and measurable also 2 The pressure–volume loops in vivo, shows muscle fibres while working; the hidden layer shows muscle fibres in their excited state before Of the four chambers that comprise the whole heart, working. We used this material model to study the the left ventricle (LV) accomplishes the major me- mechanics of the left ventricle following a top–down chanical work, while undergoing large deformations modeling approach as in (12). and intense stress states. On a simplistic level, the Within the context of our special modeling, PV ventricle is an ellipsoidal chamber, whose walls are loops may be looked at in a novel way which may in- composed of muscle fibres. It is the contraction orig- troduce new insights into the mechanism determining inated in the muscles that translates into pressure the performance of the ventricular chamber. Indeed, and/or volume changes of the chamber. PV loops are derived from pressure and volume mea- The LV cycle may be schematized as the sequence sures performed during the ; we could of four steps: filling–the diastolic phase; isovolumet- say that they are pictures of the visible state of the ric contraction; ejection–the systolic phase; isovolu-

2 metric relaxation. During the cycle, both pressure 160 and volume vary in time, and a quite useful deter- minant of the cardiac performance is the plot rep- 140 ejection resenting the pressure-volume relationship in the LV during the entire cycle, the so-called PV loop; some 120 3 of the many clues contained in the plot (see Fig. 1) are briefly summarized in the following. 100 2 Point 1 defines the end of the diastolic phase and is relaxation 80 characterized by the end–diastolic volume ved (EDV) and pressure p (EDP). At this point the mitral ed 60 valve closes and starts to contract in order to increase the . At point 2 40 contraction the systolic phase begins: the aortic valve opens and blood is ejected outside the LV. Muscles continue con- 20 4 1 tracting in order to further the ejection, while volume Left Ventricular Pressure (mmHg) diastolic filling decreases to a minimum. Point 3 defines the end of 0 the systolic phase, and is characterized by the end– 0 20 40 60 80 100 120 140 160 systolic volume ves (ESV) and pressure pes (ESP). Left Ventricular Volume (ml) Starting from here, LV undergoes an isovolumic re- laxation until point 4, where mitral valve opens and filling begins. During the filling phase, muscles con- Figure 1: Phases of the cardiac cycle of a normal tinue relaxing in order to accomodate a large increase human patient. 1) Mitral valve closes; isovolumetric in , while maintaining the pressure at a contraction. 2) Aortic valve opens; ejection. 3) Aor- quite low level. Filling is completed at point 1. The tic valve closes; relaxation. 4) Mitral valve opens, difference between maximum and minimum volume filling. The green area represents the stroke work. is called stroke volume: vstr = ved ves. Two important curves are usually− represented in a PV diagram: the end–diastolic pressure–volume re- chamber; any pressure variation will cause a volume lationship (EDPVR) and the end–systolic pressure– change along the ESPVR, provided muscles maintain volume relationship (ESPVR), see Fig. 2. These the same level of activation. Thus, the ESPVR pro- curves characterize the passive mechanical response vides an upper boundary for the pressure at which of the ventricle in two quite different states: the re- the aortic valve closes, and the position of point 3 laxed state, and the contracted one, respectively. depends on the end-systolic ejection pressure or vol- Let us consider point 1: muscles are in their most ume. relaxed state (the slack state), and any pressure vari- As it is well known, the denotes the in- ation will cause a volume change along the EDPVR, flow conditions determined by the venous system, provided the muscles stay inactive.Thus,theED- and the , the outflow conditions determined PVR provides a lower boundary for the pressure at by the arterial system. The interaction between the which the mitral valve closes, and the position of LV, the preload, and the afterload strongly influence point 1 depends on the end-diastolic filling pressure. the arterial blood pressure and the , Physiologically, the EDPVR changes as the heart which in turn constitute two key factors for the as- grows during childhood; most other changes accom- sessment of the overall cardiovascular performance. pany pathologic situations (hypertrophy, infarct, di- Thus, a change in preload or in afterload conditions lated cardiomyopathy). may markedly alter the PV loop, and results in a Let us now consider point 3: muscles are in a highly shift of point 1 or point 3 along the EDPVR or the activated state, and the LV behaves as a much stiffer ESPVR, respectively, as figure 2 shows.

3 160 Preload 2.25 140 2.20 2.15 Filling 120 3 2.10 2.05 100 2 2.00 Ejection 1.95 80 1.90

ESPVR SARCOMERE LENGTH (microns) 60 1.85 20 25 30 35 40 VOLUME (ml) 40

20 4 1 Figure 3: Sarcomere length as function of volume Left Ventricular Pressure (mmHg) (Elaboration from (16). EDPVR 0 0 20 40 60 80 100 120 140 160 Left Ventricular Volume (ml) 2.1 Elastance, compliance, and con- tractility Elastance is defined as the ratio of pressure change Figure 2: EDPVR and ESPVR represent the to volume change, provided the level of muscles acti- pressure-volume relationships for a complete relaxed vation remains the same. The notion of elastance is state, and a highly activated state, respectively. thus introduced to describe the variations of the ap- Changes in EDP make point 1 move along the ED- parent stiffness of the chamber during the cycle, and PVR; changes in ESP move point 3 along ESPVR. it is related to the actual level of activation. For example, let us consider again the two curves EDPVR and ESPVR. Both curves correspond to a A meaningful measure of the preload would proba- constant level of muscle activation, zero, and very bly be the sarcomeres length at end . Due to high, respectively. Along the first curve elastance is the intrinsic difficulties related to this measurement, clearly increasing; along the second one, it is almost EDP is the most common index of preload. Afterload constant. Compliance is simply the inverse of elas- is in general related to the arterial system, but also tance, that is, the ratio of volume change to pressure pathologic conditions, such as a leaky mitral valve or change. Thus, we could say that EDPVR is very a stenotic aortic valve, could be accounted for. compliant at low pressure and very stiffat high pres- At any point in the PV loop there corresponds a sure. specific level of muscle activation, and thus, a spe- The notion of time–varying elastance E(t) has been cific average sarcomere length. A quantitative sam- introduced to describe the pressure-volume relation- pling of sarcomere lengths versus volume is depicted ship during the entire cycle: denoted with vo arefer- in figure 3. At the preload condition sarcomeres are ence volume corresponding to zero pressure, we have slightly stretched but still not activated, then passive elasticity predominates. Isovolumic contraction and p(t)=E(t)(v(t) vo) . (2.1) ejection are the consequences of increasing muscle ac- − tivation, and thus, an increasing sarcomeres short- The idea underlying (2.1) is that of a state function ening. During isovolumic relaxation and subsequent linking three fundamental variables: for example, the filling, sarcomeres elongate, and eventually recover knowledge of the time course of elastance E(t) and their initial length. volume v(t) during the cycle allows for the predic-

4 tion of the time course of pressure. The notion of the pumping function of the heart. Still, it is diffi- elastance, introduced by Suga in the late 60s, has cult to make a one–to–one correspondence between a been a key concept for cardiac physiologists and car- therapy and its effect on the PV cycle of the heart. diologists since then; see (17) for review. In general, heart failure corresponds to a number The notion of contractility is strictly related to that of abnormalities in most of the structures involved of elastance. Indeed, contractility alters the appar- in the excitation–contraction mechanism which is at ent stiffness of muscles. For example, when positive the basis of the pumping function of the heart. Such or negative inotropic agents are administered to the abnormalities result from alterations of the shape, heart, the end systolic elastance Ees increases or de- the size, and the composition of the heart. As an creases, respectively. For this reason, Ees is consid- example, abnormal cardiac performance in patients ered to be an index of contractility. Actually, in the with chronic heart failure are caused by eccentric and clinical arena, due to the difficulty in measuring the concentric hypertrophy which are revealed by alter- ventricular volume and consequently in assessing Ees, ations of pressure–volume loops determined by differ- the ejection fraction (EF), defined as EF = vstr/ved, ent causes acting simultaneously. is preferred as an index of contractility. In the end, it seems that the interdependence of the key variables determining pressure–volume loops 2.2 Pressure–volume loops and dis- makes it difficult to establish a direct correspondence between an alteration of the pumping function of the eases heart and the direct pathology without further evi- Each pressure–volume loop is determined by the dence of the history of the patient, the electrophysio- properties of the heart by the characteristics of the logical state of the tissue and other information which and by the interplay between the PV loops do not show. heart and circulation. Pathological circumstances can act on any of these elements and alter the cor- responding PV loop with respect to the normal sta- 3 Modeling of muscle tissues tus. In (2), different pictures show the changes in EDPVR, ESPVR, and PV loop under decreased and The motor units of muscles, sarcomeres, exist at a increased contractility, increased end–diastolic vol- microscopic scale. A sarcomere contains many pro- ume, , fluid retention, and increased teinaceous filaments, the actin and the myosin. The lusitropy (that is an index of filling)1 interaction of these filaments through cross-bridges is responsible of the contractile properties of the muscle. Obviously, there is not a one–to–one correspon- Upon activation of the muscle, the actin and myosin dence between cardiac pathologies and altered PV filaments move with respect to each other causing loops. As it is explained in (2), a shift of the ES- the sarcomere to shorten. Upon de-activation of the PVR downward and to the right may correspond to muscle the actin and myosin filaments can recover an infarcted myocardium as well as to a dilated car- their original positions, pulled back by elasticity in diomyopathy. Both the pathologies are characterized the surrounding tissue. by decreased contractility which, ultimately, deter- mines the transformation of the PV loop. Here, we discuss a macroscopic model of muscle which embodies the notion of contraction (10), (12). The neurohumoral response, which has the role of This model introduces in the realm of biomechanic performing within our bodies adaptation to changes ideas developed long ago to describe some phenom- in the external world, may operate hemodynami- ena related to the change of ground states in elastic cal adjustments identifiable by alterations of the PV bodies such as plasticity or phase transitions (18). loop. The right therapies able to reverse these al- terations may have positive and negative effects on We suppose that muscle fibres are tightly embed- ded in the tissue and perform a sort of homogeniza- 1See (2), pp. 557. tion through the identification of a fibre of muscle

5 tissue with a muscle fibre. The key issue is to distin- ground state, that is, a state at zero tension. More- guish between active and passive deformations of the over, it characterizes a hidden layer as it takes up fibre. The active deformation describes the contrac- at a macroscopic level phenomena corresponding to tion of an activated and unloaded muscle fibre, that microscopic cellular processes. is, the shortening the fibre would have if left free to contract. The passive deformation measures the dif- ls l ls ference between the unloaded, contracted state, and − the visible state which is, in vivo, the sole observable state. It is worth noting that this point of view con- slack length ls lc − siders as primary the notion of contraction, instead lc l lc of that of tension: muscle tension arises whenever − contraction is hampered. contracted length 3.1 The two–layer model l σ σ We model an isolated fibre of muscle tissue, referred to from now on as the tissue specimen, as a one– visible length dimensional continuum. It may be visualized as a bar of length ls, meant to represent the muscle fibre in its slack state, that is, at zero contraction. More- Figure 4: Schematic of muscle contraction. The over, we denote with lc the contracted and unloaded tension σ developed during activation depends on length of the same fibre, and with l the length that the difference between the visible and the contracted the specimen reveals to an in–vivo observation (the length. visible length). Here, we limit our analysis to homo- geneous deformations and write From our point of view, the tension σ in the tis- sue specimen depends on the passive deformation ϕ. lc = εc ls ,l= ϕlc = ϕεc ls . (3.1) Nevertheless, the contraction εc alters the tissue re- sponse as may be easily shown through the following Figure 4 shows the relationship represented by equa- example. Letσ ˆ be the constitutive law relating the tions (3.1) through a cartoon. Equation (3.1) allows 2 passive deformation ϕ to the tension σ and let it be a peculiar reading: the total stretch represented as

ε := ϕεc (3.2) 1 σ =ˆσ(ϕ)=Yλ3(ϕ) ,λ(ϕ)= (ϕ2 1) , (3.3) 2 − of the fibre given by the ratio l/ls is multiplicatively decomposable into an active stretch εc, measuring the with Y the elastic stiffness of the tissue specimen. contraction, and a passive stretch ϕ, responsible for Equation (3.3) is a possible choice forσ ˆ which has any variation in the elastic energy. Consequently, we the advantage of being simple, and, as shall be shown require that upon activation εc < 1, and under ten- in the next section, to capture the key features of the sion ϕ>1; the slack state corresponds to εc = ϕ = 1. EDPVR and ESPVR curves. Nevertheless, different This approach is founded on a two–layer kinematic choices forσ ˆ may be done. None of them change the (8), (9), (19), (13). The total stretch ε lives at the idea underlying the model of muscle tension but just visible layer, insofar it determines the visible length the dependence of the muscle tension on the muscle of the specimen. The active stretch (for us, the con- length. The stretch ϕ is meant to depend on the traction) εc alters the length of the fibre but does actual length l and the amount of contraction εc (see not induce any tension, so that lc corresponds to a equation (3.1)2); λ is a common strain measure. For

6 l = ls, εc = 1, we have ϕ = 1 and thus σ = 0; it means that the slack length corresponds to a stress– free state. The rather simple law (3.3) yields a very compli- ant response for λ 0, which becomes quite stiff ￿ with increasing values of strain. Thus, with εc =1it captures, at least qualitatively, the passive response of cardiac muscles, and for εc < 1itdescribesthe response of an activated muscle. To summarize, the notion of contraction that we propose allows us to Figure 5: Spherical approximation of the left cham- view any curve tension versus strain as the expres- ber. sion of the same constitutive equation (3.3), which in case of muscle activation (εc < 1) stiffens. It is worth in the LV tissue is the well known Laplace formula: noting that equation (3.3) can be used to describe σh how muscles can control, independently, both posi- p =2 . (4.4) tion and tension, by allowing a change in Y and/or r εc. Actually, we are assuming that any hoop fibres in the spherical surface behave as the one–dimensional fibre studied in the previous section. The balance of the 4 Ventricular Pressure-Volume spherical surface of radius r is satisfied if equation Relationships (4.4) holds. The radius r of the sphere is related to its volume v by Here, we show how the present approach to muscle 3 v 1/3 modeling may be used in the description of the pump- v r =ˆr(v)= , (4.5) ￿→ 4 π ing function of the LV. With this aim, we shall look ￿ ￿ at pressure–volume loops from a novel point of view, so, it is easy to re–write the balance equation (4.4) and we shall introduce the notion of contraction– in terms of pressure and volume instead of pressure volume loops. and radius. The contracted volume vc corresponding The macroscopic model we present here is a zero– to the pair (p, v) is a ground volume: it is defined as dimensional model, simple enough to enlighten the the pressure–free volume of the sphere which has the key ideas at the basis of the modeling, but capable volume v under the pressure p. In particular, we as- of capturing the important features of the pumping sume that the pressure–free volume vs corresponding function of the heart which are collected in the PV to the pair (ped,ved) identifies the slack state of the loop. Moreover, it is rigorously extendible to the full chamber. Subsequently, we measure contraction from fledged non-linear 3D elasticity theory. The main there and assume that the level of muscle activation mechanical relationships we aim to discuss are not at the ground volume vc is: altered by a simplified geometry. This is the rea- son why, in the following, before turning to com- v 1/3 ε = c . (4.6) plex mathematical models for interrelating ventric- c v ular chamber pressure and volume to a suitable mea- ￿ s ￿ sure of contraction, we deal with a spherical approx- It is worth noting that, due to the balance equation imation of the left ventricle. (4.4), the pressure–free volume is also a stress–free For h the constant wall thickness of the surface, volume. Nevertheless, it is not contraction–free since a handy relation between the pressure p inside the the unique pressure–free and contraction–free volume spherical chamber and the mean tension σ generated is the slack volume vs. Moreover, we assume that

7 the visible volume v is attained from the contracted attained by εc, the ESPVR curve. Finally, solving f volume vc through an elastic deformation ϕ.Inthe with respect to εc, gives the amount of contraction following, we often refer to rc and rs as to the radii corresponding to any specific pair (p, v) corresponding to the ground contracted volume vc εc = εc(p, v; Y) . (4.12) and to the slack volume vs, respectively. Of course, rc =ˆr(vc) and rs =ˆr(vs); moreover, εc = rc/rs. Thus, the role of (4.10) is analogous to that of (2.1) The elastic strain λ of the chamber may be written and of the other empirical formulas relating pres- as sure and volume (20), that is, to subsume through 1 r λ = ( )2 1 . (4.7) macroscopic quantities the main features of PV loops. 2 r − ￿ c ￿ Nevertheless, it replaces the notion of time–varying Hence, the elastic strain λ is zero at the ground vol- elastance E(t) with that of time–varying contraction umes (r = rc) which are mechanically relaxed even if εc(t), thus expressing through a physically based pa- places of active deformation (contraction). Moreover, rameter the subtle notion of chamber contraction. In we still write the end, let us note that the assumption that the stiff-

3 ness Y does not change during contraction is just a 3 1 r 2 simplifying hypothesis which, however, does not alter σh= Yλ = Y (( ) 1) , Y = Yh, 2 rc − the capacity of the model and may be easily revised. ￿ ￿ (4.8) for the tension developed into the chamber whose 4.1 A case study slack radius is rs and denote with Y the elastic mem- brane stiffness of the chamber. Equations (4.4), (4.7), We discuss our point of view with reference to a spe- and (4.8)1 turn out a basic equation relating the pres- cific pressure–volume loop extracted by (15). There, sure p and the volume v of a spherical surface char- with reference to a normal human patient, the pairs acterized by the slack radius rs and by the stiffness pressure(mmHg)–volume(ml) are measured and the Y when contraction attains the value εc: PV loop shown in figure 6 is generated. Precisely, points from 1 to 2 describe the isovolumic contrac- 3 Y 1 r 1 2 tion; points from 2 to 3 describe the ejection phase; p =2 (( ε− ) 1) ,r=ˆr(v) . (4.9) r 2 r c − points from 3 to 4 describe the isovolumic relaxation; ￿ s ￿ and points from 4 to 1 describe the filling phase. For Let us note that, fixed rs, equation (4.9) can be a better comprehension, in figure 7, top, we represent viewed as a function f relating pressure, volume and with different colours the time course of the pressure contraction, with stiffness Y acting as a parameter: corresponding to the four characteristic phases of the cardiac cycle: blue for isovolumic contraction, red for f(p, v, ε ; )=0. (4.10) c Y ejection, green for isovolumic relaxation, and black The function f can be solved with respect to each for ventricular filling. We assume that the points la- belled 1 and 3 correspond to the end–diastolic and one of the three variables (p, v, εc), and completely characterizes the pumping action of the heart. In the end–systolic pressure–volume pairs, respectively. particular, as it will be shown, it comprises the key When fixing the value of the elastic stiffness Y of the curves EDPVR and ESPVR. Equation f, solved with chamber to 1500 mmHg cm ((21)) and defined rs as respect to p, gives a pressure–volume relationship de- the radius corresponding to the volume characteris- pending on the contraction tic of the point 1, we recover the contraction measure associated to every (p, v) state in the loop. Denoted p = p(v, εc; Y) . (4.11) with (εc)i the contraction corresponding to the state identified by the pair (pi,vi), from equation (4.12), Specifically, p = p(v, 1; Y) yields the EDPVR curve, we find and p = p(v, εc,es; Y), with εc,es the maximum value (εc)i = εc(pi,vi; Y) .

8 Prex mmHg Pressure mmHg 140

150 120 ￿ ￿ ￿ ￿ 100

80 3 60

100 40

20 2 0 Time s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Contraction 1.05 50 ￿ ￿ 1.00

0.95 4 1 0.90

0 Vol ml 0.85 50 100 150 0.80 Figure 6: A typical PV loop of a normal human pa-￿ ￿ 0.75 tient, as measured in (15), with our ESPVR and ED- 0.70 Time s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 PVR curves superimposed, as equation (4.9) dictates; the large blue dots correspond to the same four key ￿ ￿ points shown in figure 1. Figure 7: A typical pressure cycle VS time (top, mea- sured from (15)) and the corresponding contraction (bottom) as given by equation (4.9). In figure 7, bottom, the time course of the contraction cycle is represented: the end–diastolic state corre- sponds to εc = 1 and the end–systolic state to εc,es = effect is caught by the model. 0.766. Corresponding to these values, equations (4.9) The volume axis intercept v of the EDPVR gives the EDPVR and ESPVR curves which are rep- • s resented in figure 6 (blu and green solid line, respec- curve corresponds to the slack chamber at zero tively) as superimposed on the PV loop extracted stress which is determined at a ventricular pres- by (15). Moreover, the transition from EDPVR to sure of 0 mmHg. As noted in (1), this intercept ESPVR may be derived through a generalization to differs from the volume axis intercept of the ES- any point during the cardiac cycle of the procedure PVR which represents the volume of the cham- used to extract the EDPVR and the ESPVR rela- ber at zero stress and at maximum contraction. tionships from equation (4.9). The pressure–volume According to the experiments, (see figure 5 in curves corresponding to a few intermediate points in • (1)), figure 6 shows a smooth transition from the cycle are shown in figure 6 (red lines). Let us the EDPVR to ESPVR curves corresponding to note the following key points. a stiffening of the chamber due to the muscle activation. The EDPVR curve is intrinsically nonlinear, as • it is expected and as equation (4.9) rules. More- Actually, the volume axis intercepts of the over, it shows that, at subphysiological volume • pressure–volume curves at different values of ranges, increasingly negative pressures are re- contraction differ one from the other. In other quired to reduce volume. Nevertheless the sim- words, the volume axis intercept is not totally plicity of the theoretical model, a likely (see (1)) independent of the contraction state (2).

9 Figure 6 shows a non linear ESPVR. Indeed, the (identifying the slack state of the chamber) because • ESPVR is in general non linear even if reason- a contraction is acting and making the difference be- ably linear relationships are commonly used to tween states 1 and 2. characterize properties of the chamber with mus- cles in a state of activation (see equation (2.1)).

As well as a new point of view in looking at pressure– volume loops, our model allows the association of any r =3.19 r =3.16 r =2.75 r =2.78 PV loop with a contraction–volume loop. It captures 1 2 3 4 at a glance a hidden property of the left chamber and gives an idea of the contractility of the left ventricle around all the key phases of the cycle. Precisely, fig- ure 8 shows the contraction–volume loop correspond- rc,1 =2.57 rc,2 =2.31 rc,3 =1.97 rc,4 =2.27 ing to the numeric PV loop described in figure 6. It shows some key interesting features such as a vari- ability of the contraction measure in a range which Figure 9: Schematic of LV contraction. The top row is in agreement with scientific literature (2) and a shows the visible state of the LV in the four key points strong similarity to the experimental loop in figure 3. of the cycle; the bottom row shows the corresponding contracted, but unloaded, LV (actually, the first point represents the slack state). All the measures are in Contraction 1.1 cm.

1.0 1 Finally, let us introduce the deformationε ¯ with respect to the end–diastolic state; the corresponding 2 0.9 strains (¯ε 1)/2 along the cardiac cycle are shown 4 2 in figure 10− and, interestingly, are in agreement with

0.8 (22) where circumferential strains at different points 3 of the left ventricle are measured.

0.7 Volume ml 80 90 100 110 120 130 140 150 Visible Strain 0.00 ￿ ￿

Figure 8: Contraction along the cardiac cycle as func- ￿0.05 tion of volume.

￿0.10 The following cartoon sketches efficiently our point of view. Blue balls aim to represent the visible states of the left chamber in the four key points of the cycle. ￿0.15 Correspondingly, red balls represent the contracted and stress–free states. As an example, let us follow ￿0.20 Time s 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 what happens from point 1 to point 2. During the iso- volumic contraction, the visible volume of the balls ￿ ￿ is roughly the same (136ml versus 132ml), that is, Figure 10: Visible strains vs time along the cardiac red balls 1 and 2 have almost the same radius. At cycle. the same time, at a hidden layer something is occur- ring and the red ball 2 is smaller than the red ball 1

10 4.2 Ventricular–vascular coupling: ef- 160 fects of preload and afterload 140 In the previous sections, we presented our interpreta- 3￿￿ tive model of pressure–volume loops based on a dis- 120 3 tinguished notion of contraction. Here, we show further appealing characteristics of 100 3￿ our modeling, that is, its ability to predict some al- terations of the PV loop and the contraction loop, as 80 consequences of changes in preload and/or afterload ESPVR conditions. 60 As explained in (23), “if an intervention is per- formed that acutely changes the loading conditions 40 on the heart but has no effect on myocardial con- 20 1 tractility (e.g., transient inferior vena caval occlusion Left Ventricular Pressure (mmHg) to reduce preload, administration of phenylephrine EDPVR 0 to increase afterload, etc.) a family of loops is ob- 0 20 40 60 80 100 120 140 160 tained. The end-systolic and end-diastolic points of Left Ventricular Volume (ml) these loops delineate two distinct boundaries”. Let us observe that the shape of the loops is the same for every element of the family, in such a way Figure 11: The effective arterial elastance E may be that every loop can be described simply through the a represented in the PV diagram by the line connecting segment ED–ES, moving between the two rails ED- point 1 to 3, that is, ED to ES (the slope of this line is PVR and ESPVR. (see figure 11). Further, we as- actually E ). As example, any alteration of the af- sume that all the other points of the loop are scaled a terload moves− point 3 along the ESPVR, thus chang- to follow the transformation of the segment ED–ES. ing E ; the new PV loop is related to the baseline by Then, by choosing a PV loop as a reference, we can a an affine transformation (baseline is represented with reasonably describe the whole loop family through a solid line, alterations with dashed lines). two–parameter stretch transformation group. From a geometrical point of view, the simplest choice of the two key parameters is the arch-length (say (s1,s2)) to the total peripheral resistance Rp, defined as the on EDPVR and ESPVR. On the other hand, from ratio between the pressure gain pgn and the cardiac a physical point of view better choices are the pairs output (v v ) H ,withH denoting the heart ed − es R R (ped,pes), (ved,ves), and (pgn,vstr), with the pres- rate: sure gain pgn := pes ped. Actually, we introduce − pes ped pgn Ea a further parameter which quantifies the properties R = − = = . (4.14) p (v v ) H v H H of the arterial system, the effective arterial elastance ed − es R str R R Ea defined by Equation (4.14) quantifies the ventricular-vascular coupling, as it relates , stroke volume, and pes ped pgn Ea = − = . (4.13) peripheral resistance. For example, it shows that al- ved ves vstr − tered peripheral resistance results in a variation of The importance of Ea is twofold: it admits a straight- the heart rate and/or the effective arterial elastance, forward graphical representation on the PV diagram that is, of the points ED, ES. As shown in (1) a useful which highlights the role of the two key points 1 (ED) parametrization of the curve family can be (ved,Ea), and 3 (ES), see Fig. (11). In fact, Ea represents the in the sense that we can choose (ved,Ea) and obtain, slope of the segment ED–ES and is strictly related for example,ves.

11 In the following, with the aim of showing the typ- Prex mmHg ical alterations of the PV loops due to changes in 150 preload and/or afterload conditions, we will use the ￿ ￿ transformation group based on the two parameters (ped,pes). Thus, the altered PV loop may be recon- 3 structed as follows: let us consider pressure as inde- pendent variable, and the function (4.10) solved with 100 respect to the volume: v = v(p, εc; Y ); changes in 2 2 preload and/or afterload implies that ped p¯ed = p + p and/or p p¯ = p + p ￿→,respec- ed ￿ ed es ￿→ es es ￿ es tively; accordingly, volumes ved and/or ves change as 50 follows 4 1

ved v¯ed = v(¯ped,εc,ed; Y) , 1 ￿→ (4.15) 4 ves v¯es = v(¯pes,εc,es; Y) . 0 Vol ml ￿→ 50 100 150

Thus, under the assumption that εc,ed, εc,es, and Y ￿ ￿ stay constant, points ED and/or ES move along the Figure 12: Changes in the baseline PV loop (red line, EDPVR and/or ESPVR curves, respectively. More- solid) due to an increase and a decrease of the preload over, any other point (¯pi, v¯i) in the altered PV loop (black line, dashed). may be given in terms of the baseline points (pi,vi) through the affine transformation 5 Geometrical and material re- p¯i p¯ed =¯pgn/pgn (pi ped) , − − (4.16) modeling at a glance v¯i v¯ed =¯vstr/vstr (vi ved) . − − In the previous sections, we presented our inter- Eventually, equation (4.10), solved with respect to pretative model of pressure–volume loops based on a distinguished notion of contraction which allows contraction, εc = εc(p, v; Y ), yields the new values of a macroscopic modeling of the pumping action of contractionε ¯ci needed to sustain the altered condi- tions the heart. Pressure–volume loops are viewed as de- termined by three main state variables, pressure, ε¯ci = εc(¯pi, v¯i; Y) . (4.17) volume, and contraction related through equation (4.10). Actually, other variables determine the char- These values, a prediction of the model, may shed acteristics of PV loops. A few of them depend on light on the amount of the physiological compensa- the circulatory system and are represented in the tion needed as the consequence of altered preload pressure–volume plane by the notion of preload and and/or afterload: in order to pump blood effectively afterload discussed in the last section. Other vari- under the new conditions, that is, to fulfill the con- ables depend on the heart structure; we think of the dition that PV loops transform affinely, muscle con- size and the stiffness of the left chamber and refer to tractions have to change. the ability of the heart tissue to shorten and relax. Figure (12) shows the effects of altered preload on All these elements strongly vary in the presence of the baseline PV loop shown in figure 6; next figure heart failure which, in its turn, may be revealed just (13) shows the associated alterations in the contrac- through the analysis of the corresponding pressure– tion loop as predicted by (4.17). Figure (14) and (15) volume loops. show the same effects for altered afterload. Here, we aim to discuss as an independent variation

12 Contraction Prex mmHg 1.2 150 ￿ ￿ 1.1 3

1.0 1 1

0.9 2 100 3 4 2 4 2 0.8 3 2 0.7 Volume ml 60 80 100 120 140 160 50

￿ ￿ Figure 13: Alterations in the baseline contraction– 4 4 1 volume loop (red line, solid) corresponding to the in- crease and decrease of the preload (blue line, dashed). 0 Vol ml 50 100 150

￿ ￿ of the size, the stiffness, and the ability to contract Figure 14: Changes in the baseline PV loop (red line, of the left chamber, represented within our model solid) due to an increase and a decrease of the after- through the triplet (vs, Y,εc), allows to recover some load (blue line, dashed). typical alterations which pressure–volume loops show in presence of heart failure. Even if applied here in a simple context, the basic idea is that a well–founded write equation (4.10) as model of the pumping function of the heart allows the intervention on specific parameters and gives back f(p, v, εc(rs); Y)=0, (5.18) meaningful information. So, in a patient–specific analysis in the presence of altered pressure–volume in such a way to make evident the role of the ra- loops, the model may suggest the role of different pa- dius rs as reference radius for the contraction mea- rameters in determining the alteration and the right sure. Equation (5.18), solved with respect to v gives tool for recovering the normal situation. back the volumes vi corresponding to the pressures pi: vi = v(pi,εc(rs); Y). In the presence of a size al- teration, we have a new reference radius αm rs and, Change in size corresponding to the same pressures, new volumes vi∗ as We denoted with vs the volume axis intercept of v∗ = v(pi,εc(αm rs); Y) . (5.19) the pressure–volume curve corresponding to the end– i diastolic state which views the chamber as relaxed The altered PV loop (dashed lines) corresponding to and stress–free. Moreover, we remember that,in our the old pressures and to the remodeling parameter modeling, vs is the reference volume for the measure αm =1.04 is shown in figure 16 together with the of contraction as equation (4.6) says. We model size normal PV loop (solid lines). The changes in EDPVR abnormalities just as alteration of the relaxed and and ESPVR show as the enlarged chamber induces stress–free size of the chamber through the scalar re- a decreased cardiac efficiency. Indeed, the change in modeling parameter αm such that αm rs measures the EPVR curve says that at the same pressure a the radius of the altered chamber. For αm < 1 and larger volume of the chamber is attained. Moreover, αm > 1 a reduction and an enlargement of the left an increase of the stroke volume turns out when en- chamber is accounted for, respectively. So, let us largement occurs while pressures stay normal.

13 Contraction Prex mmHg 1.2 150 1.1 ￿ ￿

1.0 1 3 3

2 0.9 4 2 100 4

0.8 2 2 3 3 0.7 Volume ml 60 80 100 120 140 160 50

￿ ￿ Figure 15: Visible strain vs time along the cardiac 4 4 1 1 cycle. 0 Vol ml 50 100 150 5.0.1 Change in contractility ￿ ￿ The ability to contract and relax of the left cham- Figure 16: The altered PV loop in presence of remod- ber is accounted for through the parameter εc.We eling (αm > 1 (dashed lines) versus the normal loop note that equation (4.12) allows any pressure–volume (solid lines). pair in a PV loop to associate with the correspond- ing contraction εc. Hence, a variation of εc means a variation of the ability of the chamber to contract at 5.0.2 Change in stiffness any time along the cardiac cycle. Of course, it em- In conclusion, we discuss the role of the stiffness of bodies a variation of the maximum contraction ε c,es the chamber in the modeling through the analysis and, consequently, a variation of the ESPVR curve. of the consequences of a variation in the chamber We account for such a variation through the scalar stiffness. Firstly, let us note that here Y is a global contractility parameter α so that c measure of stiffness and it does not allow to account for the stiffening of portions of the left ventricle due εc + αc (1 εc) (5.20) − to the replacement of the normal tissue by scar such measures the altered ability to contract. Still, equa- as after a large myocardial infarction. Nevertheless, tion (4.10), solved with respect to v gives back the the typical degradation of the elastic properties of the new volumes corresponding to the normal pressures chamber occurring in heart failure may be accounted when the contraction of the chamber is uniformly al- for through uniform decrease of the chamber stiffness tered: Y. Equation (4.10), solved with respect to v gives back the volumes corresponding to an altered stiffness

vi∗ = v(pi,εc + αc (εc 1); Y) . (5.21) of the chamber when pressures and contractions stay − unchanged: For αc < 0 and αc > 0 a decrease and an increase in εc is produced, respectively. Figure 17 shows as an vi∗ = v(pi,εc; αs Y) ,αs > 0; (5.22) increase in contractility corresponding to αc =0.4is revealed by the stiffening of the ESPVR curve, which αs is the key element in the parametric analysis. Fig- is related to the contraction measure at the point 3. ure 18 shows as pressure–volume loops change with Of course, the EDPVR curve stays unchanged as it respect to the normal one when a strong decrease in corresponds to a contraction measure εc = 1. the chamber stiffness is accounted for.

14 Prex mmHg Prex mmHg

150 150 ￿ ￿ ￿ ￿

3 3 3 3

100 100

2 2 2 2

50 50

4 4 1 4 4 1 1

0 Vol ml 0 Vol ml 50 100 150 50 100 150

￿ ￿ ￿ ￿ Figure 17: The altered PV loop in presence of a pos- Figure 18: The altered PV loop in presence of a de- itive variation in contractility (αc > 0 (dashed lines) crease in the chamber stiffness (αs < 1 (dashed lines) versus the normal loop (solid lines). versus the normal loop (solid lines).

References

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