Connecting developed pressure – Preload relationship in ex-vivo beating heart with cellular sarcomere length – Tension relationship

Abstract

The heart’s developed pressure (DP) in Langendorff heart experiments increases with preload via the Frank-Starling mechanism up to a critical transition point at which DP starts to decrease with preload. A similar behavior is found at the cellular level, where the tension developed by skinned cardiac fibers or myocytes in isometric tension test increases with sarcomere length up to a transition point beyond which, the tension decreases. This cellular-level behavior is termed myofilament length dependent activation. While these two behaviors are similar, they occur at vastly different scales. Specifically, the DP – preload and sarcomere length – tension relationships occur, respectively, at the organ and cellular scales. Correspondingly, it remains unclear how much these behaviors are related. To address this issue, we use computer modeling that connects cellular to organ mechanics found in the ex-vivo beating rat heart experiments to determine whether the DP – preload relationship at the organ level can be explained solely by the sarcomere length – tension relationship at the cellular level. We found that the non-monotonic behavior of the DP with preload is consistent with a model predicted feature of myocardial contractility. The LV sarcomere length at the transition where DP and myocardial contractility start to reduce is 2.12 ± 0.03 $\mathrm{\mu }\mathrm{m}$. This transition sarcomere length is outside the range of 2.2 – 2.4 $\mathrm{\mu }\mathrm{m}$ that is associated with the peak tension found in skinned rat cardiac fibers or myocytes with isometric tension test. This disparity suggests the presence of other factors affecting the DP – preload relationship found in Langendorff heart experiments such as the prescribed initial length of sarcomere that vary between different rat species.

1 Introduction

The Langendorff setup is used to understand fundamental heart physiology and diseases ( ^{Bell et al., 2011}) by measuring the heart’s contraction under pathophysiological conditions. The Frank-Starling mechanism is reflected in the Langendorff system by an increased developed pressure (DP) of the left ventricle (LV) under isovolumetric condition in response to an increased preload. The increased DP with preload holds up to a critical transition beyond which, the DP decreases with increasing preload ( ^{Goshovska et al., 2022; Piuhola et al., 2003; Takahashi et al., 2019}). This organ-level behavior mirrors the cellular-level behavior found in isometric tension experiments conducted on skinned cardiac fibers or isolated cells, where peak tension is increased with sarcomere length up to a transition sarcomere length ( ^{Ter Keurs et al., 1980}). This cellular-level behavior is referred as myofilament length dependent activation ( ^{Kentish et al., 1986}), and the transition sarcomere length is found to be between 2.2 and 2.4 $\mathrm{\mu }\mathrm{m}$ experimentally. The similarity between organ level DP – preload and the cellular level tension – sarcomere length relationships has led to the conclusion that the non-monotonic behavior of the DP – preload relationship is explained by the myofilament length dependent activation behavior. The Langendorff experiments and the skinned muscle isometric tension test are different. There are other factors contributing to these distinct behaviors. As such, it remains unclear whether observations at the organ level are attributable to those found at the cellular level ( ^{de Tombe and ter Keurs, 2016}). To resolve this issue, we combined Langendorff experiments and computational modeling ( ^{Fan et al., 2023; Fan et al., 2021a; Fan et al., 2021b}) to test the hypothesis of whether the organ level DP – preload relationship can be explained by the cellular level tension – sarcomere length relationship.

2 Methods
2.1 Animals

Animal experiments approval using male Sprague-Dawley rat weighting 300–350 g was granted by the Institutional Animal Care and Use Committee of Michigan State University. All animal procedures were conducted in accordance with the Guiding Principles in the Care and Use of Animals.

2.2 Ex-vivo beating heart Langendorff experiments

Animals (n = 5) were deeply anesthetized with 4–5 % isoflurane after 15 mins of heparin injection (100 U/kg, i.p.). The animals were decapitated after they became unresponsive to noxious stimuli, and a thoracotomy was performed. The heart was excised immediately from the animals and submerged in cold Krebs-Henseleit bicarbonate (KH) buffer. The aorta was rapidly attached and tied to an aortic cannula that is connected to 80 mmHg perfusion line, mounted on a Langendorff perfusion system (Radnoti, US), perfused via the Langendorff mode of constant flow at a flow rate of 10 ml/min with oxygenated KH buffer at 37 ${}^{°}$C and pH 7.4.

The KH perfusate is equilibrated with 95/5% O ₂/CO ₂ gas mixture and filtered through a 0.22 $\mathrm{\mu m}$ membrane filter prior to use and continuously filtered with an in-line filter connected to the perfusion system.

The heart was paced at 4 Hz by a pair of electrodes placed on the right atrium. Capture was verified by changing the pacing rate and the resultant mechanical activity (> 99%) correlated with the pacing stimuli. A size 6 latex balloon was inserted into the LV via the pulmonary vein. Baseline was established by inflating the balloon by injecting warm saline solution until the LV end-diastolic pressure (LVEDP) is between 5 – 20 mmHg. The heart was stabilized for 15 – 20 mins. The balloon was inflated incrementally to increase preload until the LVEDP is 60 mmHg. After the heart was stabilized for each preload that is representative of the cavity volume here, the LV and aortic pressures were each continuously recorded by a pressure transducer connected to the isovolumetric beating heart ( ^{Fig. 1} a). The rate pressure product that is the product of heart rate and systolic blood pressure, temperature, and ECG were continuously recorded (16 Channels Powerlab, AD Instrument) in LabChart Pro. After the Langendorff experiment, the heart was dissected to measure the length, basal inner and outer diameters, and wall thickness of the LV. Each measurement was performed three times and averaged to obtain dimensions for the model.

Data analysis of the left ventricle.

Left ventricle pressure waveforms, which were acquired after the heart was stabilized at each preload, were averaged over 10 cardiac cycles in the analysis of measurements. Left ventricle end-diastolic pressure–volume relationship (EDPVR) and end-systolic pressure–volume relationship (ESPVR) were constructed by connecting the EDP and ESP at each preload, respectively.

2.3 Finite element formulation of the left ventricle mechanics

A prolate ellipsoid LV geometry was constructed based on the geometrical measurements ( ^{Fig. 1} b). A LV finite-element (FE) model with myofiber helix angle varying transmurally from −60° at the epicardium to 60° at the endocardium ( ^{Fig. 1} c) was developed to simulate isovolumetric contraction corresponding to the Langendorff experiment ( ^{Bell et al., 2011}). The LV FE model is implemented using the open-source library FEniCS ( ^{Fan et al., 2023; Fan et al., 2021a; Fan et al., 2021b; Mojumder et al., 2023}).

Constitutive law of the left ventricle

The stress tensor of the LV P is decomposed additively into a passive ${\mathit{P}}_{\mathit{p}}$ and an active component ${\mathit{P}}_{\mathit{a}}$ (i.e., $\mathit{P}={\mathit{P}}_{\mathit{p}}+{\mathit{P}}_{\mathit{a}})$. The passive stress tensor is defined by ${\mathit{P}}_{\mathit{p}}=dW/d\mathit{F}$, where F is the deformation gradient tensor and W is a strain energy function of a Fung-type transversely-isotropic hyperelastic material ( ^{Guccione et al., 1991}) ( ^{Appendix A}). The active stress ${\mathit{P}}_{\mathit{a}}$ is calculated along the local fiber direction using the active constitutive relationship ( ^{Guccione et al., 1993}) ( ^{Appendix A}).

2.4 Simulation cases and postprocessing

Isovolumetric contraction with different preloads corresponding to the experiments is simulated to fit the measured EDPVR and ESPVR. To fit the measured EDPVR, passive parameters C, b _ff, b _fx and b _xx were calibrated using the modified Levenberg-Marquardt algorithm ( ^{Fan, 2012}) until the root mean square error of the LVEDP is less than 5 mmHg. Starting with an initial guess of passive parameters, C was first calibrated, with b _ff, b _fx and b _xx fixed, until the model predicted EDP agrees with the measurements at the lowest preload. Specifically, the parameter C is recursively updated by multiplying it with the ratio of the measured to the model predicted EDP at the lowest preload. Next, the exponent parameters b _ff, b _fx and b _xx were calibrated to fit the EDPVR while maintaining the ratio of these parameters ( ^{Fan et al., 2023; Fan et al., 2021a; Fan et al., 2021b}). The values of C, b _ff, b _fx and b _xx for each case are reported in ^{Table B1} of ^{Appendix B}. To fit the ESPVR, the LV contractility $T_{\mathit{max}}$ was adjusted separately for each preload so the model predicted ESP agrees with the measurements within an error of 5 mmHg.

At each preload, the stretched sarcomere length of the myofiber and contractility were computed based on the length-dependent calcium sensitivity ${\mathit{EC}}_{a50}$. When the sarcomere is stretched, it becomes more sensitive to the calcium concentration therefore develops more force. The Frank-Starling mechanism is primarily driven by the length-dependent ${\mathit{EC}}_{a50}$ ( ^{Tanner et al., 2023}) but also controlled by the other components i.e., the binding of actin and myosin ( ^{Park-Holohan et al., 2021; Tanner et al., 2023}). The time-averaged maximum fiber strain over the LV is used to determine the sarcomere length.

3 Results

The measured developed pressure (DP) at baseline is greater than 50 mmHg in all the hearts ( ^{Fig. 2} ).

When preload increases, EDP and ESP increase from 0 to above 65 mmHg and from close to 0 to above 100 mmHg, respectively, at the highest preload ( ^{Fig. 3} ). Model predicted LV pressure waveforms agree well with the experimental measurements at different preloads in the 5 cases. The average error between the model prediction and measurement is 20% and the maximum error occurs mainly during diastole at higher preload in case 4. The model predicted EDPVR and ESPVR are in agreement with the experimental measurements.

The measured EDP increases exponentially with an increase in preload represented by the LV volume in the 5 cases ( ^{Fig. 4} a). The EDP, on average, increases by 12 mmHg over 5 cases when the LV volume is less than 280 $\mathrm{\mu L}$ but the average increase becomes 56 mmHg when the LV volume is greater than 280 $\mathrm{\mu }\mathrm{L}$. Compared to EDP, the measured ESP increases with preload at constant rate ( ^{Fig. 4} b). The DP, which is the difference between ESP and EDP, increases first then decreases when preload increases. This transition occurs when the balloon volume is between 280-340 $\mathrm{\mu m}$ with the maximum DP varying between 50–80 mmHg. The computed sarcomere length at the transition point varies between 2.10 to 2.16 $\mathrm{\mu m}$ in the 5 cases, with a mean value ± standard deviation as 2.12 ± 0.03 $\mathrm{\mu m}$ ( ^{Fig. 4} c). The trend of model predicted contractility-preload relationship is consistent with the DP-preload relationship ( ^{Fig. 4} d).

4 Discussion

This study integrates LV FE modeling with Langendorff beating heart experiments to provide insights on the association of DP generated by whole heart at the organ level to sarcomere length and active tension generated at the cellular level. The LV DP reflects the overall pressure output, which is determined by the preload, afterload and contractility. Myocardial contractility measures the intrinsic ability of myocardial fibers to generate force, which is load-independent. The overall finding of this study is that the non-monotonic behavior of DP-preload is consistent with the myocardial contractility estimated from the computational model predicted active tension. The average transition LV sarcomere length at which DP and myocardial contractility start to reduce is 2.12 ± 0.03 $\mathrm{\mu m}$ (n = 5). The estimated transition sarcomere length of 2.12 ± 0.03 $\mathrm{\mu m}$ is below the range of 2.2 – 2.4 $\mathrm{\mu m}$ corresponding to the highest force generated by rat’s myofiber found in isometric tension experiments ( ^{Ter Keurs et al., 1980; Weiwad et al., 2000}).

Left ventricle developed pressure – preload relationship found here agrees with previous experiments ( ^{Goshovska et al., 2022; Piuhola et al., 2003}). Specifically, a previous study ( ^{Goshovska et al., 2022}) shows that with an increased balloon volume from 0 to 235 $\mathrm{\mu }\mathrm{L}$, the DP increases from 55 to a maximum value of 130 mmHg at a balloon volume 100 $\mathrm{\mu }\mathrm{L}$, following which the DP decreases. Although the magnitude of DP and balloon volume are different from our measurements, the non-monotonic trend is consistent where DP increases first then reduces when preload increases. These disparities may be attributed to the different rat species (Wistar rats) used previously ( ^{Goshovska et al., 2022}). The non-monotonic DP-preload behavior found in another study ( ^{Piuhola et al., 2003}) uses 7–8 week old male Sprague-Dawley rats, where the magnitude of DP is comparable with our measurements ( ^{Fig. 4} b) but the critical LV volume is smaller than ours as we used older rats (∼ 12 weeks).

Myocardial contractility (indexed by the active tension), at different preloads, increases first then decreases above the critical balloon volume. This behavior is similar to the sarcomere length-tension behavior found in isometric tension test using rats’ isolated myocytes or skinned fibers ( ^{Guccione et al., 1997; Weiwad et al., 2000}) where the transition sarcomere length at peak isometric tension is between 2.2 – 2.4 $\mathrm{\mu m}$. The model predicted sarcomere length range 2.10 – 2.16 $\mathrm{\mu }\mathrm{m}$ at which DP peaks before decreasing is below those found in the isometric tension experiments ( ^{Fig. 4} c). The difference in peak active tension/contractility may be attributed to the different conditions, where the sarcomere-length tension was measured on isolated myocytes from epicardium and the myocardial contractility was averaged over the whole LV. Besides, measurements were taken when the sarcomere was fixed but the sarcomere length is simulated in dynamic condition.

5 Conclusion

In this study, we found that the change of the DP in response to the increased preload (LV volume) at the organ level is consistent with the changes of the contractility in response to the increased preload at the cellular level, where both DP and contractility increase first then reduce with increased LV preload in the Frank-Starling relationship. We found that the sarcomere length corresponding to the peak DP in the Langendorff experiments of 2.12 ± 0.03 $\mathrm{\mu m}$ is outside the range of sarcomere length corresponding to the peak tension found in isometric tension experiments (2.2 – 2.4 $\mathrm{\mu }\mathrm{m}$). These results suggest other factors affect the DP – preload relationship such as the prescribed initial sarcomere length that varies between different rat species.

CRediT authorship contribution statement

Lei Fan: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Vahid Ziaei-Rad: Writing – review & editing, Visualization, Validation, Methodology, Formal analysis. Jason Bazil: Writing – review & editing, Supervision, Methodology, Funding acquisition, Conceptualization. Lik Chuan Lee: Writing – review & editing, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the National Institute of Health ( R01 HL134841 and R01 HL163977), NSF ( 2222066) and American Heart Association Postdoctoral Fellowship ( AHA 835298).

Appendix A Passive and active stresses

Fung-type transversely-isotropic hyperelastic material is given by (1) $W=\frac{1}{2}C\left(e^Q-1\right)$In Eq. ⁽¹⁾, (2) $Q=b_{\mathit{ff}}{E}_{\mathit{ff}}^2+b_{\mathit{xx}}\left(E_{\mathit{ss}}^2+E_{\mathit{nn}}^2+E_{\mathit{sn}}^2+E_{\mathit{ns}}^2\right)+b_{\mathit{fx}}\left(E_{\mathit{fn}}^2+E_{\mathit{nf}}^2+E_{\mathit{fs}}^2+E_{\mathit{sf}}^2\right)$where E _ij with ( i, j) ∈ ( f, s, n) are components of the Green-Lagrange strain tensor $\mathit{E}$ with f, s, n denoting the myocardial fiber, sheet, and sheet normal directions, respectively. Material parameters of the passive constitutive model are denoted by $C$, $b_{\mathit{ff}}$, $b_{\mathit{xx}}$, and $b_{\mathit{fx}}$. The parameters $C$ and b ( $b_{\mathit{ff}}$, $b_{\mathit{xx}}$, and $b_{\mathit{fx}}$) reflect the stiffness of myocardium and exponential behavior of the heart tissue, respectively, and they vary regionally. The parameters $b_{\mathit{ff}}$, $b_{\mathit{xx}}$, and $b_{\mathit{fx}}$ represent anisotropy when their values are different.

The active stress is given as (3) ${\mathit{P}}_{\mathit{a}}=T_{\mathit{max}}\frac{C_{a0}^2}{1+E{C}_{a50}^2{(E}_{\mathit{ff}})}\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}(\omega (t,t_{\mathit{init}}))}{2}{\mathbf{e}}_f⨂{\mathbf{e}}_{f_0}$where $C_{a0}$ is the peak intracellular calcium concentration, $T_{\mathit{max}}$ is myocardial contractility, and given by (4) ${\mathit{EC}}_{a50}=\frac{{{(C}_{a0})}_{\mathit{max}}}{\sqrt{\exp \left(B\left(l-l_0\right)\right)-1}}$

In Eq. ⁽⁴⁾, $B$ is a material constant, ${{(C}_{a0})}_{\mathit{max}}$ is a prescribed maximum peak intracellular calcium concentration, $l_0$ is sarcomere length at which no active tension develops, $l$ is instantaneous sarcomere length based on prescribed initial length of sarcomere $l_{s0}$ and the fiber strain $E_{\mathit{ff}}$ as (5) $l=l_{s0}\sqrt{2E_{\mathit{ff}}+1}$and the wave function $\omega (t,t_{\mathit{init}})$ is (6) $\omega \left(t,t_{\mathit{init}}\right)=\left\{\begin{array}{c}\frac{\pi t}{t_0},0\le t<t_{\mathit{init}}\\ \frac{\pi t_{\mathit{init}}}{t_0},t\ge t_{\mathit{init}}\end{array}\right.$where $t_0$ is the time taken to reach peak tension and $t_{\mathit{init}}$ is the transition time. In Eq. ⁽⁵⁾, the sarcomere length in the unloaded reference configuration $l_{s0}$ is prescribed to be 1.95 $\mu m$ based on measurements (Bub et al., 2010). The fiber strain $E_{\mathit{ff}}$ is related to the right Cauchy-Green deformation tensor $\mathit{C}$ and the unit vector in the myofiber direction in the reference configuration ${\mathit{e}}_{\mathit{f}}$ by (7) $E_{\mathit{ff}}=\frac{1}{2}({\mathit{e}}_{\mathit{f}}\bullet \mathit{C}\bullet {\mathit{e}}_{\mathit{f}}-1)$

Appendix B Passive parameters

The values of passive parameters C, b _ff, b _fx and b _xx used in each case are reported in ^{Table B1}.

1

Values of passive parameters used in each case.

Case Number	$\mathrm{C}$(k Pa)	${\mathrm{b}}_{\mathrm{f}\mathrm{f}}$	${\mathrm{b}}_{\mathrm{f}\mathrm{x}}$	${\mathrm{b}}_{\mathrm{x}\mathrm{x}}$
1	34.92	39.62	18.17	36.60
2	27.24	50.00	19.72	39.72
3	161.85	29.42	13.49	27.18
4	7.64	47.75	21.90	44.11
5	5.27	36.61	16.78	33.82

The values of C vary from 5.27 to 161.85 kPa, values of b _ff vary from 29.42 to 50.00, b _fx vary from 13.49 to 21.90, and b _xx vary from 27.18 to 44.11 over 5 cases.

Appendix C Limitations

There are some limitations associated with this study. First, an idealized LV geometry is used in this study. In future, we will reconstruct the subject-specific LV geometry based on 3D ECHO images acquired from rat hearts in-vivo. Second, we assumed the same transmural variation of the myofiber helix angle for all the cases. In future work, subject-specific geometries and myofiber orientation measured from diffusion tensor magnetic resonance imaging can be used to increase the model’s realism.