Citation:Kato S, Himeno Y, Amano A (2024) Mathematical analysis of left ventricular elastance with respect to afterload change during ejection phase. PLoS Comput Biol 20(4): e1011974. https://doi.org/10.1371/journal.pcbi.1011974
Editor:Alison Marsden, Stanford University, UNITED STATES
Received:September 20, 2023; Accepted:March 7, 2024; Published: April 18, 2024
Copyright: © 2024 Kato et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability:The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting information files including program code of the model.
Funding:The author(s) received no specific funding for this work.;
Competing interests: The authors have declared that no competing interests exist.
Introduction
The primary function of the heart is to transport arterial blood to the various organs and venous blood to the lungs by the ventricular myocyte contractile force. Since the pump function is impaired in the pathological conditions, quantitative indices which represents the heart contractility was interested by clinical cardiologists which may become an early indication of myocardial disease [1]. The pump function of the heart has been measured by many indices such as cardiac output, stroke volume, peak systolic pressure, and rate of rise of left ventricular pressure. However, since there are muscle cell characteristics that the contraction force increases with its length (force length relation), and decreases with its velocity (force velocity relation), measurements of these indices are affected by the preload which determines the cellular length at end-diastole, and afterload which determines the cellular contraction velocity during ejection phase [2]. Thus a single universal quantitative description of the cardiac pump function was investigated in 1970s, and Suga and Sagawa has proposed Emax as an index of cardiac pump function which is not affected by the cardiac load conditions.
Emax is a well-known index of cardiac contractility, which is the slope of the end-systolic pressure-volume relation (ESPVR), where ESPVR is obtained from the end-systolic points of several pressure-volume loops (PV loops) by using different preloads or afterloads on the left ventricle (LV) [3, 4]. The word Emax represents the maximum elastance of LV, and the concept that it represents the contractility of LV is based on the assumption that the LV elastance increases during the ejection phase and reaches its maximum value at the end-systole. Emax is considered a good index of LV contractility as it was considered to be load independent.
There are many researches on Emax. For example, several studies have found that Emax increases with adrenergic stimulation [4–8], and decreases in ailing conditions such as heart failure [9–11].
However, in several studies, Emax was found to be afterload-dependent [12] and ESPVR to not be linear but convex [13, 14]. Moreover, it was found that ESPVR’s volume intercept will move with load [15, 52]. These points are briefly summarized in [16].
Emax is experimentally obtained by gradual preload reduction [17]. Since this approach is not easy to apply in the clinical situation, methods to estimate Emax have been suggested by using a single PV loop [11]. However, there is still no good method to estimate Emax, limiting its applicability in clinical practice.
These limitations suggest that the theoretical basis of the Emax is still not clear and requires further investigation.
In the related reports, Emax are measured by the slope of pressure and volume relation at end-systole. By generalizing this elastance to the arbitrary time point during ejection phase, we can think of the elastance as the slope of pressure and volume relation at given time, and this elastance is represented as “load-dependent elastance” (Eload) in this paper, where several pressure and volume points with different LV loads at a given time are necessary to determine it.
On the other hand, as we can consider LV as a compartment with pressure and volume, from the amount of the pressure change caused by the instantaneous volume change, the physical elastance of LV can be defined by the ratio between the pressure and volume change. This elastance can be recognized as a physical elastance property at a given time. Thus, in this paper this elastance is represented as “instantaneous elastance” (Einst).
Historically, Templeton et al. measured the ratio between the pressure and the volume change by applying the 22 Hz sinusoidal volume change to canine LV [18]. From our definition, we can consider this ratio to be close to Einst. Additionally, they reported that this value was linearly related to pressure throughout the cardiac cycle. However, the temporal change in the ratio was not clearly reported.
Technically, it is very difficult to directly measure LV elastance. There are several reports on the measurement of cardiac tissue elastances [19–22]. Saeki et al. measured kitten’s papillary muscle stiffness by applying an oscillation of 0.1 to 60 Hz and reported that stiffness changes with the cardiac cycle, but the amplitude of the difference was not large [23].
Historically, instantaneous elastance and load–dependent elastance were conceptually considered to represent similar properties of LV. However, there are very few reports on the measurement of instantaneous elastance; thus, it is difficult to compare these two. We now have cellular contraction models based on molecular level findings capable of reproducing muscle kinetic properties. Therefore, in this paper we try to analyze the characteristics of instantaneous elastance and load–dependent elastance based on the molecular level muscle contraction model by using the hemodynamic model combined with the contraction model.
Model and definitions
Model structure
In this research, a simplified hemodynamic model proposed by our group [24, 25] was used for mathematical analysis. The model was constructed from a circulation model, a LV geometry model, and a muscle contraction model.
The variables used in the model equations are summarized in Table 1.
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Table 1. Variables of the simplified circulation model.
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Circulation model
Because our aim was to mathematically analyze the pressure and volume relation of the model, we used a simplified circulation model based on the Windkessel model (Fig 1). The parameters in the model were basically imported from the circulation model proposed by Heldt et al. [26] and Liang et al. [27].
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Fig 1. Circulation model.
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The model included pulmonary venous pressure (PE1), LV pressure (Plv), aortic pressure (Pa), and peripheral pressure (PE2) as pressure variables. For simplicity, we used constant values for PE1 and PE2. The model also included pulmonary venous resistance (Rpv), aortic resistance (Rlo), and peripheral resistance (Rout) as vascular resistance parameters. The LV volume was denoted by Vlv and the aortic volume was denoted by Va. The flow between compartments were denoted as q0 for the LV incoming blood flow, qin for the aortic blood flow, and qout for the peripheral blood flow. The aortic compliance was denoted by Ca, which has a relation with the aortic pressure and volume as follows.(1)
As our aim was to reproduce the baseline hemodynamics, we did not include the baroreflex effect in the model. Moreover, we fixed the cycle length at 1000 [ms]. To evaluate the effect of changes in the afterload, we used several values for Rout. The parameters used in the circulation model are shown in Table 2.
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Table 2. Parameters of the circulation model.
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LV geometric model
As our analysis was based on the molecular-level contraction model, a LV geometry model was necessary to relate LV pressure and volume with cellular level contraction force and sarcomere length. We used a measurement-based fitting function that relates 1) the LV internal radius (Rlv) to Vlv, and 2) Rlv with half-sarcomere length (L), respectively. Also, Laplace’s law [28] was used to relate the LV wall tension (Fext) with LV pressure (Plv).
* Relation between the LV volume and internal radius
Utaki et al. [24, 25] defined the equation for the relation between Rlv and Vlv using the following reported data. Corsi et al. [29] measured the time course of the human LV volume, and Sutton et al. [30] measured the time course of the human LV internal radius. Combining these data, a non-linear relation between the LV volume and the internal radius was obtained. However, as the resolution of these data was insufficient, they used the time course of the canine LV volume reported by Rodriguez et al. [31] and the time course of the canine LV internal diameter reported by Sabbah et al. [32] to draw a non-linear relation between the LV volume and internal radius. Given that these are canine data, scaling to human data was performed, and the non-linear equation between the LV volume and internal radius was obtained (Fig 2 nonlinear curve). In this study, to simplify the mathematical analysis, we approximated the non-linear equation between the LV volume and internal radius into the linear equation as follows (Fig 2 linear line).(2)Here, KR and KV are constants adjusted to the physiological relation between Rlv and Vlv during end-diastole to end-systole.
* Relation between the LV myocardial sarcomere length and internal radius
Utaki et al. [24, 25] used the following reported data to define the relation between Rlv and L. Rodriguez et al. [31] measured the time course of canine LV sarcomere length, and Sabbah et al. measured the time course of the canine LV internal diameter [32]. By combining these two, a non-linear relation between the LV volume and sarcomere length was obtained (Fig 3). Also it was assumed that the characteristics were basically similar with canines and humans, thus the scaling factor to the relation was introduced. In this study, we used the same equation shown in Eq (3) that Utaki et al. [24] used (Fig 3 dotted line).(3)
Here, KL and Lb are constants.
* Simplified LV wall thickness equation
LV wall thickness is known to become maximal at the end-systole and minimal at the end-diastole. In a recent report, not only the LV volume but also the LV twist angle was found to be related to the wall thickness [33]. Thus, wall thickness is not always proportional to the LV volume [29, 30], and the quantitative mechanism of wall thickness remains unclear. Utaki et al. [24] assumed that wall thickness is linearly related to the cellular contraction force. However, in this study, to simplify the mathematical analysis, we used constant wall thickness during ejection phase.(4)Here, hlv [cm] denotes the LV wall thickness. Utaki et al. [24] assumed hlv = 1.00 [cm] at the end-diastole and hlv = 1.70 [cm] at the end-systole; thus, we used hlvED = 1.35 [cm] as the constant wall thickness in our analysis.
* Laplace’s law
As Laplace’s law represents well the relation between Plv, hlv, Fext, and Rlv [34], we used this relation in our model.(5)Here, Fext [mN/mm2] denotes the LV wall tension. The calculation method of Fext is explained in section Muscle cell contraction model. Ku is a parameter that converts units from [mN/mm2] to [mmHg].
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Fig 2. Relation between Rlv and Vlv obtained from the data reported by Rodriguez et al. [31] and Sabbah et al. [32] (non-linear), and linearized function used in the model (linear).
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Fig 3. Relation between LV radius Rlv and half sarcomere length L obtained from data by Rodriguez [31] and Sabbah [32], and approximated linear relation (dotted line).
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To reproduce the physiological hemodynamic parameters, we used the model parameter values shown in Table 3.
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Table 3. Parameters of the LV geometry model.
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Muscle cell contraction model
In this study, we used a muscle cell contraction model proposed by Negroni and Lascano (NL08 model) [35]. NL08 model consists of chemical state transition model shown in Fig 4 and mechanical model which represents elongating spring which has sliding connection point shown in Fig 5. In the model, the cellular contraction force (Fb [mN/mm2]) is calculated by multiplying the probability of the power generation states of the troponin system and crossbridge length (hp [μm], hw [μm]).(6) (7) (8) (9) (10) Aw and Ap represent the spring rate of crossbridges in the weak (w) and power (p) states that generate Fb. [] [μM], [TS∼] [μM], [] [μM] and [TS*] [μM] represent the concentrations of the troponin system forming the crossbridges in the weak (∼) and the power (*) states. As shown in the following equations, changes in these states are calculated by the rate constants shown in Fig 4, where the Ca2+ bound troponin system state TSCa3 increases from free troponin TS with increase in Ca2+ concentration, and crossbridge forming state TSCa3∼ increases with rate constant f.(11) (12) (13) (14) (15)
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Fig 4. Chemical state transition and rate constants of NL08 model.
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Fig 5. Mechanical model of NL08 model.
Contraction force is determined by the spring rate determined by the chemical state model multiplied by the crossbridge length (hp, hw). Note that the crossbridge consists of weak (w) and power (p) states.
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Note that to realize the cooperativity characteristics, three troponins are handled as one troponin system in the model. On the other hand, crossbridge forming state TSCa3∼ and TS∼ decreases to crossbridge unbound states TSCa3 and TS by the rate constant g and gd whose values increases with crossbridge extension hw − hwr. X [μm] is the non-elastic portion of the contractile element and L − Xw and L − Xp are extensions of the attached crossbridges in the weak and power states, abbreviated as hw and hp (Eqs (7) and (8)), respectively. The time derivative of Xp and Xw are given by the products of the crossbridge sliding rate (B) and the crossbridge extension (hp − hpr, hw − hwr), which are different from the initial length(hpr[μm], hwr[μm]) shown in Eqs (9) and (10). Note that [], [TS∼], [], [TS*], hp, and hw are calculated using the equations in the original paper of the NL08 model [35] with modifications of g and gd to be explained later.
The contraction time course is controlled by Ca2+, and Ca2+ release is controlled by the Ca2+ release equation. The release and absorption of Ca2+ by the sarcoplasmic reticulum (Qrel [μM/ms] and Qpump [μM/ms]), and also the changes in the Ca2+ in the NL08 model are expressed by the following equations.(16) (17) (18) (19)
Note that, t [ms] is the time parameter, [Ca2+] [μM] is the concentration of Ca2+, Qm [μM/ms] is the maximum level of Ca2+ release, t1 [ms] is the interval to maximum Qrel, Qpump_rest [μM/ms] determines [Ca2+] at rest, Kp [μM/ms] is the maximum value of Qpump, and Km [μM] is the value of [Ca2+] for Qpump = Kp/2. Itroponin represents the amount of Ca2+ bounds to the troponin system. Parameter values used in Eqs (16) and (17) are shown in Table 4.
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Table 4. Parameters used in Eqs (16) and (17).
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The NL08 model is known to have a problem in the ejection phase, where Fb rapidly decreases when L shortens. As shown in Fig 4, g and gd are the rate constants which determines the decreasing rate of crossbridges. Since these values increase with (hw − hwr)2, TSCa3 and TSCa3* decrease when increase which happens at both ejection and filling phase. To improve the filling phase characteristics, Kγ was introduced in [24], and we used this modification in our model as follows.(20) (21) (22)
Note that, hwr [μm] is the steady state extension of the attached crossbridges in the weak state. Za [1/ms] and Yd [1/ms] are crossbridge dissociation constants, Yv [1/ms] and Yvd [1/ms] are model parameters for the weakly-attached crossbridge extension, and Yc [1/ms/μm2], and Lc [μm] are model parameters related to the half-sarcomere length.
As the characteristics of the end-diastolic pressure-volume relation (EDPVR) are similar in rats [36] and humans [37], by linearly scaling the force axis with the identical half-sarcomere length axis, we used the following mammalian exponential function as a human passive elastic force (Fp [mN/mm2]) model showing good agreement with the experimental data [38, 39]. The form of this equation was based on the equation used by Shim et al. [40] and Landesberg et al. [41].(23)
Note that L0 [μm] is resting half sarcomere length. D, KPL [mN/mm2], and KPE [mN/mm2] are constant parameters that determine the properties of the passive elastic component. The parameter values were manually adjusted to reproduce physiological human hemodynamics (Tables 5 and 6).
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Table 5. Parameters in Eq (23).
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Since Fp is usually measured using a piece of tissue or an LV cavity, we can consider that the characteristics of Fp are compatible with the macroscopic properties. On the other hand, since Fb is usually measured with a single cell or a small piece of ventricular fiber in which the effective cross-sectional area is difficult to measure, the measured force may contain large scale errors. We thus introduced a scale factor, KS, which is multiplied only with Fb to adjust the cellular contraction force. KS was determined using the method proposed by Utaki et al. [24], which resulted in KS = 7.24. Finally, LV wall tension Fext [mN/mm2] in Eq (5) was calculated as follows.(24)
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Table 6. Parameters used to calculate cellular contraction force (Fb).
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Mathematical definitions of elastance
* instantaneous elastance
Considering LV as one elastic compartment, the instantaneous elastance Einst can be defined by the ratio between the instantaneous changes in pressure (Plv) and small changes in volume (Vlv), as can be defined as follows.(25)
* load-dependent elastance
For the heart, pressure and the volume follow a different pressure-volume curve if the afterload is different. Usually, the end-systolic pressure volume curve is measured by measuring several pressure-volume curves with different loads; thus, we can define load-dependent elastance Eload by the ratio between the changes in pressure with changes in afterload and volume changes with the changes in the afterload as follows.(26)
Simulation
Simulation conditions
Simulation program was written in C, and performed in 0.01 [ms] intervals until a periodic limit cycle was achieved, requiring around 100 cardiac cycles. In this study, during the simulation of 100 cardiac cycle to achive periodic limit cycle and following measurement of elastance, when Plv exceeded 70.0 [mmHg] during the isovolumic contraction phase, Pa and Va were fixed to constant values of 70.0 [mmHg] and 105.0 [mL], respectively, which corresponds to the onset condition of ejection phase. In addition, Plv after the onset time of ejection rapidly changed to 70.0 [mmHg], and Vlv also rapidly changed to 126.419 [mL]. We also measured the Einst by decreasing the Vlv by 0.25% (ΔVlv) at a certain time in the ejection phase and by evaluating the pressure drop (ΔPlv) in the next time interval. Thus, Einst was calculated by ΔPlv/ΔVlv.
Simulation results
To evaluate the influence of the afterload on pressure (Plv) and volume (Vlv), we performed a simulation with several peripheral resistance values: Rout = 1050, 1200, 1400, 1650, and 2050 [mmHg ⋅ ms/mL]. Under these conditions, the PV loops are shown in Fig 6 and the enlarged ejection phase in Fig 7. In both figures, the isochronous points of Plv—Vlv are shown at 50, 100, 150, 200, 250, and 280 [ms] after the onset time of the ejection, and the isochronous linear approximation lines of each time point are also shown in these figures. The values of each slope (Eload) are shown in Fig 8, and Eload, Vlv-axis intercepts (V0), and R2 of these lines are shown in Table 7. From the results, the isochronous lines had high R2 values, thus the isochronous points of Plv—Vlv had high linearity during the ejection phase. We also noted that V0 greatly decreased with time, but after the initial drop, Eload slightly decreased over time, although the amount of decrease was quite small as we can see in Fig 7. With the instantaneous volume drop simulation, we also measured Einst (Fig 9) at the same time points shown in Fig 8 with Rout = 1050, 1200, 1400, 1650, 2050. As you can find in Fig 9, Einst slightly increased and decreased during the ejection phase.
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Fig 6. PVloops and isochronous Plv—Vlv relations.
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Fig 7. Isochronous Plv—Vlv relations during the ejection phase.
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Fig 8. Changes in Eload with time.
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Fig 9. Changes in Einst with time.
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Table 7. Changes in Einst, Eload, V0 with time, and its R2.
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In the model, the relation between volume (Vlv) and half-sarcomere length (L) is linear, as shown in Eqs (2) and (3). On the other hand, although the relation between LV pressure (Plv) and wall tension (Fext) is nonlinear, as shown in Eq (5), it can be approximated as linear for a small change of Plv. Thus, if we convert the isochronous relation of Plv—Vlv in Fig 7 into the isochronous relation of Fext—L shown in Fig 10, the relation remains linear. The linear approximations for the isochronous points of Fext—L have high-linearity, as evidenced by R2 in Table 8. The slopes (ka) of these lines slightly decreased, but they appear to be almost parallel.
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Fig 10. Isochronous Fext—L relation during the ejection phase.
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Table 8. Changes in the slope of the isochronous Fext—L relation (ka) with time and its R2.
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Results
Elastance of simplified hemodynamic model
From the simulation results, it can be seen that Einst slightly increased and then decreased, and its time course was similar to that of [Ca2+]. On the other hand, Eload was markedly different from Einst at each time point. Eload initially decreased by around 20%, and decreased slightly over time. On the other hand, if we measure Einst under different Rout (Fig 9), we can find that the values at the same time points are very close. For the measurement of Eload, we have to change Rout, however, from Fig 7, we can find that the isochronous PV points lie on a line, that means that the slope is constant and does not depend on Rout, which is similar with the characteristics of Emax which is considered to be a good index of contractility because of its independence with respect to afterload. Thus we can say that at a certain time point, both elastances were independent of the afterload (Rout). The mathematical reasons for these findings are discussed in the following section.
Einst in the simplified hemodynamic model
Einst was mathematically defined by Eq (25), and by using this model, we can represent Einst as follows.(27)
From Eq (5), the partial derivative can be derived as follows.(28)
Derivative can be derived from Eq (2) as follows.(29)
Similarly, from Eq (5), the partial derivative can be derived as follows.(30)
From Eqs (2) and (3) and (24), the derivative can be derived as follows.(31)
The elastance of the cellular contraction force term can be derived by differentiating Eq (6) with L as follows.(32)
Also, by differentiating Eq (23) with L, can be derived as follows.(33)
Although Eqs (32) and (33) appear in Eq (31), from the simulation results, was about 100 times larger than . Thus, we can neglect the term in Eq (31).
In this model, Einst is denoted by Eq (27). From the simulation results, we found that the second term of the right-hand side of Eq (27) was dominant. The second term of the right-hand side of Eq (27) is the product of Eqs (30) and (31), that represents the changes in the LV pressure by the changes in the LV wall tension, and the changes in the LV wall tension by the changes in the LV volume, respectively. Since term was dominant in Eq (31) and is considered to be proportional to the attached cross bridges from Eq (32), finally, Einst can be approximated by the following equation (Eq (34)).(34)
In this model, increases with , and increases with [TSCa3], and [TSCa3] increases with [TS] × [Ca2+]. Thus, as we can find from the changes in these normalized concentrations shown in Fig 11, tends to follow changes in [Ca2+]. And since [TS*] is small, Einst can be understood as the LV wall tension elastance mainly determined by the Ca2+ concentration.
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Fig 11. Normalized concentration of [Ca2+], [TSCa3], , and Einst.
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Eload in the simplified hemodynamic model
Eload was mathematically defined as Eq (26). Although the simulation conditions were not physiological, we fixed Plv at the onset time of ejection to simplify the mathematical analysis. The phenomena in the ejection phase can be divided into two steps: 1) initial phase and 2) steady state transition during the ejection phase.
1. step 1 Immediately after the onset time of ejection, transient variations in crossbridge extension changes occur in short time periods, depending on the afterload.
2. step 2 The linear isochrones of the Plv—Vlv relations remain linear during the ejection phase with small changes in their slopes.
step 1.
It is known that the response to transient force is observed in the initial phase of isovelocity-shortening experiments using skeletal muscles. However, the phenomenon is difficult to measure under accurate isovelocity conditions in real experiments [42–45]. At this phase, the contraction force changes significantly depending on the velocity. In our analysis, as Plv at the onset time of ejection was fixed, the shortening velocity of the half-sarcomere length () becomes roughly proportional to the inverse of the afterload (Rout), corresponding to the quasi isovelocity-shortening condition. Thus, we can predict that the variation of the initial cellular contraction force resulting from the variations of the afterload is caused by the same mechanism.
We performed an isovelocity-shortening simulation by using the NL08 model under the condition of the onset time of ejection. Since the effect of variations in Rout to [Ca2+] is small, [Ca2+] at the time of ejection can be considered as constant for this short period, even if the afterload (Rout) is different. Thus, we fixed [Ca2+] as 0.3 [μM], and performed an isovelocity-shortening simulation of the half-sarcomere length () with shortening velocities of 0.1, 0.15, and 0.20 [μm/s] corresponding to the condition close to the hemodynamics simulation afterloads. The time courses of L, hp, and Fb are shown in Figs 12, 13 and 14, respectively. Note that the isovelocity condition starts at t = 100 [ms] in these figures.
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Fig 12. Time course of half-sarcomere length (L).
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Fig 13. Time course of the crossbridge length (hp).
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Fig 14. Time course of the cellular contraction force (Fb).
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As shown in Figs 13 and 14, crossbridge length (hp) of the power state becomes constant after an initial decrease and the time course of Fb is similar to that of hp.
From the model equations, the shortening velocity of the half-sarcomere length () was found to be inversely proportional to the afterload (Rout). Thus, hp changes depending on Eqs (8) and (9)as follows.(35)
After the initial decrease of hp, it exponentially got close to the steady state length hp,ss, which can be calculated from as follows.(36)
From this equation, hp decreased from the resting crossbridge length (hpr) to the steady state length that is linearly related with the shortening velocity of the half-sarcomere length (). This is what produces the well-known characteristics of the force-velocity relation. Thus, if the troponin system probabilities are constant, the cellular contraction force (Fb) is proportional to hp, which is also proportional to . As a result, the isochronous Fext—L (≈Fb—L) points located at the same position of the onset time of ejection, move in the L direction, depending on the velocity () determined by the afterload. However, the movements done in the Fb direction are maintained by the relation between and , which can be approximated with a linear relation as shown in Eq (35). Thus, the initial cellular contraction force and length relation becomes almost linear.
If we approximate the initial transient changes in and for duration Δt as linear with respect to time, by denoting at time Δt after onset of ejection as , the changes of L and Fext, ΔL and ΔFext can be approximated as follows.(37) (38)
Then, the initial slope between Plv and Vlv which can be considered as initial Eload is approximated by the following eqution.(39) (40)
In the simulation results in section Simulation results, the shortening velocity of the half-sarcomere length () was affected by the cellular contraction force (Fb); thus, its ratio was slightly different from the inverse of the afterload (Rout). However, the difference between hp and hp,ss was around 2.3% at 15 [ms] after the onset time of ejection.
step 2.
As explained in the previous section, the fixed Fext—L points with different afterloads (i.e. fixed Plv—Vlv points) proceeded to isochrones immediately after the onset time of ejection.
We assume that the relation between Fext and L is linear, that is Fext = kaL+ kb at a certain time t0 during the ejection phase. If we assume that linearity is maintained after the small period (Δt), then the following equation holds.(41)
In the time derivative form, the following equation holds.(42)
Here, we introduce the assumption that changes in the slope of ka from the Fext—L relation are small and can be neglected in a short time period, i.e., ; then, the following equation holds.(43)
If we assume that the increase in Fext has term proportional to L as follow, the resulting Fext—L relation still becomes linear with time.(44)
Thus, if the model equations can be decomposed into the above equation form, we can say that the model itself has in its nature the property of a linear Fext—L relation.
First, by temporally differentiating the LV wall tension (Eq (24)), we get as follows.(45) (46) (47) (48) (49) (50) (51)
In Eq (51), if L < L0, then is proportional to . If L ≥ L0, then is a function of L and . However, from the simulation results, changes in L with Rout is 0.63%. Thus, we approximated , and the passive elastic force component is approximated by using Eq (44).
On the other hand, in Eq (46), is composed of four terms. If we look at these terms, T1 and T2 are similar to T3 and T4, respectively; thus, the mathematical analysis of T1 and T2 becomes the same for T3 and T4, respectively. Moreover, Ap is five times larger than Aw. Therefore, we can reduce our analysis of Eq (46) to the analysis of T3 and T4.
T3 is a product of the crossbridge length (hp) and the time derivative term of . As explained in section step 1, hp decreases from the resting crossbridge length (hpr), and the amount of decrease is proportional to the shortening velocity of the half-sarcomere length . However, as we can see from Fig 13, the decrease of hp is so small that, hp can be approximated by hpr. In addition, the time derivative of is affected by L and . The effect of L comes from Eq (11) where the term can be approximated to be linear with L in the short time period. In general, hp can be approximated to hpr and then Fb becomes proportional to . As the time derivative of is approximated to be proportional to the value itself, T3 can be approximated to be linear with Fb.
T4 is a product of and the time derivative of hp. As mentioned above, the power state () can be approximated to be linear with Fb. The time derivative of hp can be deformed as follows.(52)
As explained in section step 1, except for the initial ∼15 [ms] after the onset time of ejection, Eq (52) reaches its steady state. This means that can be approximated as a small constant. As the value of Fp is under 2% and increases to 8% of Fext during the ejection phase, we approximate Fb as Fext. Afterward, by using K1, K2 and K3 as constants, the following equation holds.(53)
Afterward, by using Eqs (3) and (2), the time derivative of the half-sarcomere length () can be derived as follows.(54)
Here, the sum of the LV volume (Vlv) and the aortic volume (Va) during the ejection phase is denoted as the total volume (Vtot).(55)
Subsequently, can be represented as follows.(56)
During the ejection phase, the time derivative of the total volume () corresponds to the blood flow from the aorta to the periphery, which can be represented as follows.(57)
Moreover, as aortic resistance (Rlo) is small, the following approximation holds.(58)
Now, , which is the first term on the right-hand side of Eq (56), can be approximated as follows.(59)
In addition, from Eqs (1), (55) and (58), the total volume (Vtot) can be approximated as follows.(60)
From Eq (60), in the first term on the right-hand side of Eq (56) can be obtained as follows.(61)
Similarly, by differentiating Eq (60) with the LV wall tension (Fext), that appears in the second term on the right-hand side of Eq (56) can be obtained as follows.(62)
Here, the first term on the right-hand side of Eq (56) is the product of Eqs (61) and (59). Since in Eq (61) becomes around -0.07, the changes in with respect to changes in Fext can be neglected. While Eq (59) consists of the component proportional to Fext and constant. Thus the first term on the right-hand side of Eq (56) can be approximated as follows.(63)
On the other hand, the second term on the right-hand side of Eq (56) is the product of Eq (62) and . As explained above, since in Eq (62) can be neglected, and becomes constant with respect to Fext, this term can be approximated as proportional to , as changes in other terms are small with respect to changes in Fext and L. Thus the second term of right-hand side of Eq (56) can be approximated as follows.(64)
In summary, the following approximation holds with the constants K4, K5, and K6.(65)
By using this equation and Eq (53), the following equation holds.(66)
Therefore, except for the initial transient period, the relation between and approximately satisfies Eq (44) during the ejection phase. This occurs under the assumption that the relation between L and Fext satisfies Fext = kaL + kb.
In the simulation results at 50 and 100 [ms] after the onset time of ejection, we can find a highly linear relation between and , as shown in Fig 15. Similarly, at 280 [ms] after the onset time of ejection, we also can find a highly linear relation as shown in Fig 16.
[Figure omitted. See PDF.]
Fig 15. The relation between and at 50, 100 [ms] after the onset time of ejection.
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[Figure omitted. See PDF.]
Fig 16. The relation between and at 280 [ms] after the onset time of ejection.
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Thus, from the above analysis, we can understand why the isochronous LV wall tension (Fext)—half-sarcomere length (L) relation remains linear under the variation of the afterload (Rout) during the ejection phase; that is, the characteristics come from the combination of the characteristics of the cardiac muscle and compliance of the LV and aorta.
Elastance of the time-varying elastance model (TVEM)
The Time Varying Elastance Model (TVEM) is one of the major simple LV model often used in hemodynamic models. In the model, a time-dependent elastic function (E(t)) and the constant zero-pressure fill volume V0 are used to relate LV pressure and LV volume as follows.(67)
Here, we used the following function for E(t) proposed by Heldt et al. [26].(68)
Note that, Ed, Es represents LV elastance at end-diastole and end-systole, respectively. represents cardiac cycle length and duration of LV activation calculated by Bazett equation [46].
From the above equation, we can find that in the TVEM, Einst and Eload always acquire the same value of E(t).
Discussion
Physical and mathematical interpretation of Einst and Eload
As explained in the Introduction, if we consider LV as an elastic compartment, the physical elastance of LV becomes Einst which represents the transient ratio between the compartment pressure and the volume. The mathematical analysis of Einst shown in section Einst in the simplified hemodynamic model revealed that Einst is directly related with the cellular contraction force, affected by the LV radius by the effect of Laplace’s law. As the cellular contraction force is generated by Ca2+ concentration with certain delay from Ca2+, Einst becomes almost constant during the ejection phase, as shown in Fig 9.
On the other hand, Eload is derived from the concept of the time varying elastance model which came from the concept of Emax. The concept of the time varying elastance model is based on the assumption that Vlv at Plv = 0 is always same, thus at the same cardiac phase time, the LV physical compartment elastance becomes the same, thus the pressure and the volume points lie on the same straight line. However, if we use the circulation model with detailed cellular contraction model, the pressure and the volume points lied on the line whose volume intercept move with time, which is different from the time varying elastance concept (Figs 6 and 7).
Mathematically, Eload is defined by Eq (26), which has similar representation with the analysis of the stability of the limit cycle in the bifurcation analysis. Thus, we can understand that Eload corresponds to the movement of the periodic limit cycle with changes in Rout. And the difference between the values of Einst and Eload can be understood from this consideration. The analysis of the slope of Eload in section Eload in the simplified hemodynamic model clarified that the slope itself comes from the initial force velocity relation, where the initial velocity is determined by the rate of volume change related with Rout. The latter analysis clarified that the slope is kept almost linear during the ejection phase, which finally results in the linear ESPVR.
From the above consideration, the index which directly represent the elastance of LV is Einst, however in the clinical situation, the important index of the LV function is the resulting hemodynamics which corresponds to the periodic limit cycle of the LV pressure and the volume. This may be the reason that Emax which is Eload at end-systole is accepted as the good index of LV contractility.
Relation between Emax and Eload
Emax is the slope of ESPVR proposed as an index of cardiac contractility [3, 47]. From the simulation results, the value of Eload at 280 [ms] after the onset time of the ejection, which was close to that of the end-systole, was close to Emax. With our hemodynamic model, the end-systolic time slightly changed with respect to the afterload (Rout); thus, the isochronous Fext—L relation at a fixed time close to the end-systole time became slightly different from the ESPVR. However, from the simulation results, the end-systolic time was almost linear with the LV end-systolic pressure (Plv); thus, the end-systolic points were in a line and the difference between the line and the isochrones of Fext—L was very small.
Relation between force-velocity relation and Eload
In our analysis, the initial isochronous Plv—Vlv relation slope was determined by the initial transient changes in the half-sarcomere length and cellular contraction force, as explained in section step 1. Moreover, the slope was approximately maintained during the ejection phase. In this period, the sarcomere shortening velocity was determined by the mechanical load to the cell corresponding to the force-velocity relation (FVR); thus, the slope, i.e., Eload was strongly related to the FVR characteristics but not to the cardiac tissue contractility.
Since the shape of FVR is known to be physiologically non-linear [48–50], the isochronous PV relation should become non-linear. However, as the range of the force variation is quite small, FVR becomes approximately linear, resulting in a linear isochronous PV relation.
On the other hand, the positive inotropic effect, such as the noradrenergic stimulation, increases Emax, which corresponds to the increase in Eload. This can be explained by the changes in FVR under the positive inotropic effect. As reported in Sonnenblick et al. [48–50], the positive inotropic effect changes the properties of FVR. In our model, we did not include the inotropic effect. Thus, this aspect must be considered in the future model.
Time course of Einst and Eload
From Table 7, we can observe that Einst slightly increases and then decreases, while Eload monotonically decreases during the ejection phase. As shown in section Einst in the simplified hemodynamic model, Einst directly reflects cellular contractility. Thus, the time course of Einst is similar to the cellular contraction force (Fb). Additionally, as shown in section Eload in the simplified hemodynamic model, Eload reflects the characteristics of FVR. Therefore, the time course of Eload is considered to reflect the time course of the ratio between and .
Comparison between the simplified hemodynamic model and TVEM
Using a model in which the TVEM is used instead of the simplified hemodynamic model of the LV compartment, which was used by Heldt et al. [26], we evaluated the PV loops under the following peripheral resistance conditions: Rout = 1050, 1200, 1400, 1650, and 2050 [mmHg ⋅ ms/mL]. The resulting PV loops and isochronous of the Plv—Vlv relations at 5, 50, 100, 150, and 200[ms] after the onset time of ejection are shown in Fig 17. Moreover, the time course of elastance (E) is shown in Fig 18. As shown in Fig 17, the slope of the isochronous Plv—Vlv relation monotonically increases. In addition, as shown in Fig 18, E also increases monotonically. This is determined by the elastance function used for TVEM. As explained in section Simulation results, the time course of the Eload of our simplified hemodynamic model slightly decreases with time, which is different from that of the TVEM. Note that, the elastance function of TVEM varies with researchers, thus the time course of elastance changes with each research, however, since the function is constructed from the changes in the physiological pressure and volume, basic characteristics becomes similar.
For the animal experiment, Nishioka et al. [51] measured the isochronous PV relation in dogs and the results came close to our simulation model. That is, the slope of the isochronous PV relation did not change with time, but the volume intercept changed. However, in their experiment, pressure and volume at the onset time of ejection were not fixed. Thus, we cannot say that their results correspond to those of our simulation, but there are some similarities with our results.
[Figure omitted. See PDF.]
Fig 17. Isochronous Plv—Vlv relations during the ejection phase when using TVEM.
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[Figure omitted. See PDF.]
Fig 18. Time course of elastance (E) after the onset time of ejection when using TVEM.
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Influence of simplification of the hemodynamic model
To simplify the mathematical analysis, we simplified the hemodynamic model proposed by Utaki et al. [24, 25] for two points. First, LV wall thickness was fixed at a constant length, and second, the relation between the LV internal radius (Rlv) and LV volume (Vlv) was simplified to linear. However, these simplifications may reduce the accuracy of the model.
Since the contribution of the LV wall thickning to the LV pressure is considered to be not small, the effect of fixing LV wall thickness (hlv) to the hemodynamics is not small. Simulation results of both models are shown in Fig 19, where the LV pressure becomes lower in the hlv fixed model. However, the characteristics of the linear ESPVR is still preserved in the hlv fixed model.
[Figure omitted. See PDF.]
Fig 19. Simulation results using LV wall thickness linear with cellular contraction force Fb (hlv Fb linear) used in the original Utaki model [24], and fixed value (const hlv) used in the proposed simplified model.
Note that, Rout = 1050, 1400, 2050 were used in the simulation.
https://doi.org/10.1371/journal.pcbi.1011974.g019
If we use the original hlv, which is proportional to the cellular contraction force (Fb), the Einst equation (Eq (27)) becomes as follows. (69)
Comparing Eq (27) with Eq (69), we can see that the third term on the right-hand side of Eq (69) is ignored in our analysis. This term may lower the accuracy of the simulation results; however, from the simulation results shown in Fig 6, the simplified hemodynamic model still can reproduce physiological hemodynamics.
As shown in Eq (2), the relation between the LV radius Rlv and the volume (Vlv) is simplified to a linear relation in our model as shown in Fig 2. This simplification may affect the resulting pressure and volume time course. To validate that the simplification can reproduce physiological LV characteristics, the pressure and the volume time course with both the linear and the nonlinear relation were simulated in the condition without fixing Plv and Vlv at the onset time of ejection (Fig 20). The results showed not small difference between these models, however, the time courses of the LV pressure and volume for both models are in the physiological range, so that the simulation results and the analysis results can be considered to be the results within the physiological conditions.
[Figure omitted. See PDF.]
Fig 20. Simulation results using linear and nonlinear Rlv and Vlv relations.
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Additionally, as shown in Eq (54), the relation between the half-sarcomere length (L) and the LV volume (Vlv) is simplified to a linear relation in our model, while the relation becomes non-linear in dogs, as reported by Rodriguez et al. [31]. In this case, the isochronous PV relation may become non-linear. In the case of rats and sheep, the isochronous PV relation is reported as non-linear [52, 53]. Thus, part of the difference with our simulation results may be caused by this simplification.
Limitations of the analysis
In the analysis of step 2 (section step 2), the slope of the relation between the LV wall tension (Fext) and half-sarcomere length (L) is assumed to be fixed, while as shown in the simulation results (Table 7), the slope slightly decreases during the ejection phase. We could not derive an analytical solution without a fixed slope value; thus, in this aspect, we need further analysis of the mathematical equations of the model.
In the calculation of from Eq (56), and are calculated from Eqs (61) and (62). Note that, these equations cannot be calculated when .
Relation with the 3D models
Computational modeling of cardiac function have now progressed to multi-scale modeling realizing 3D geometry coupled with cellular functions and circulating fluid dynamics [54–58], and reproducing pathological conditions [59].
These models are intended to use detailed elements in the whole model to reproduce accurate behavior of the heart, however, since the large deformation elastic model calculation often becomes unstable, the elements used in these models are often simplified to reproduce physiological results [60]. Thus the reproduced aortic pressure, flow velocity or ventricular motion of these models have fair similarities with the clinical data, however, detailed analysis on the relation between the hemodynamics and the microscopic molecular dynamics are often difficult, and also the mathematical analysis of the dynamics of these models are often impossible. On the other hand, as shown in the results, the simplified lumped models of LV and circulation system have disadvantages in reproduction accuracy, however, such models are still useful since they are capable of being mathematically analyzed for their dynamics.
Conclusions
In this paper, we evaluated the simulation results of the proposed hemodynamic model that included a detailed cellular contraction model. By defining two different LV elastances: 1) instantaneous elastance and 2) load-dependent elastance, we were able to evaluate these elastances from the simulation model and found that these elastances showed markedly different time courses. That is, the instantaneous elastances showed a bell-shaped curve corresponding to the cellular contraction force, while the load-dependent elastance hardly changed with time. We then analyzed the mechanism that determines these elastances from the model equations, and found that the instantaneous elastance directly coincided with the cellular contraction force, while the load-dependent elastance was determined by the characteristics of both the instantaneous isovelocity-shortening and the force-velocity relation of cardiac cells. The slope of the isochronous pressure-volume relation was mainly determined by the velocity-dependent force drop characteristics in the instantaneous shortening. Moreover, the linear relation between the isochronous pressure and volume was based on the characteristics that the relation between the time derivative of the cellular contraction force and the cellular shortening velocity becomes linear, which comes from the combined characteristics of the LV and aortic compliances.
Supporting information
S1 Model Source Code. Contains C program code of the model.
https://doi.org/10.1371/journal.pcbi.1011974.s001
(ZIP)
S1 Model Equation. Contains model equations.
https://doi.org/10.1371/journal.pcbi.1011974.s002
(PDF)
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Citation: Kato S, Himeno Y, Amano A (2024) Mathematical analysis of left ventricular elastance with respect to afterload change during ejection phase. PLoS Comput Biol 20(4): e1011974. https://doi.org/10.1371/journal.pcbi.1011974
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About the Authors:
Shiro Kato
Roles: Formal analysis, Methodology, Software, Writing – original draft
Affiliation: Department of Bioinformatics, Ritsumeikan University, Kusatsu, Shiga, Japan
Yukiko Himeno
Roles: Validation, Writing – review & editing
Affiliation: Department of Bioinformatics, Ritsumeikan University, Kusatsu, Shiga, Japan
Akira Amano
Roles: Conceptualization, Formal analysis, Methodology, Project administration, Supervision, Writing – original draft
* E-mail: [email protected]
Affiliation: Department of Bioinformatics, Ritsumeikan University, Kusatsu, Shiga, Japan
https://orcid.org/0000-0002-7704-2458
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Abstract
Since the left ventricle (LV) has pressure (Plv) and volume (Vlv), we can define LV elastance from the ratio between Plv and Vlv, termed as “instantaneous elastance.” On the other hand, end-systolic elastance (Emax) is known to be a good index of LV contractility, which is measured by the slope of several end-systolic Plv—Vlv points obtained by using different loads. The word Emax originates from the assumption that LV elastance increases during the ejection phase and attains its maximum at the end-systole. From this concept, we can define another elastance determined by the slope of isochronous Plv—Vlv points, that is Plv—Vlv points at a certain time after the ejection onset time by using different loads. We refer to this elastance as “load-dependent elastance.”
To reveal the relation between these two elastances, we used a hemodynamic model that included a detailed ventricular myocyte contraction model. From the simulation results, we found that the isochronous Plv—Vlv points lay in one line and that the line slope corresponding to the load-dependent elastance slightly decreased during the ejection phase, which is quite different from the instantaneous elastance.
Subsequently, we analyzed the mechanism determining these elastances from the model equations. We found that instantaneous elastance is directly related to contraction force generated by the ventricular myocyte, but the load-dependent elastance is determined by two factors: one is the transient characteristics of the cardiac cell, i.e., the velocity–dependent force drops characteristics in instantaneous shortening. The other is the force–velocity relation of the cardiac cell. We also found that the linear isochronous pressure–volume relation is based on the approximately linear relation between the time derivative of the cellular contraction force and the cellular shortening velocity that results from the combined characteristics of LV and aortic compliances.
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