Evaluating Cardiac Function in Ex Vivo Heart Perfusion Using Lumped Parameter Models

Abstract

Background:

Heart transplant outcomes and survival depend on the ability to implant well-functioning organs, but there remain no reliable, objective measures of cardiac function prior to implantation. The lack of standardized protocols and advanced technologies results in inconsistencies and subjective assessments, increasing the risk for postoperative graft dysfunction, the leading cause of short-term morbidity and mortality after transplant. Ex-vivo heart perfusion (EVHP) provides a platform to evaluate donor hearts prior to implantation, using machine perfusion to reanimate the heart to a beating, physiologic state. The FDA-approved Organ Care System (OCS) is widely utilized for the evaluation and ex vivo preservation of hearts, particularly from donors after circulatory death (DCD). However, it does not permit a physiological assessment of heart function because, while the heart continues to beat, its chambers remain devoid of perfusate and thus are unable to perform any functional work.

Method:

In this study, we developed and validated a lumped parameter mathematical model to assess donor hearts during ex-vivo perfusion, using a customized, in-house EVHP setup that allows left ventricular loading.

Results:

We demonstrate the ability of our mathematical model to accurately predict hemodynamic parameters, enabling performance analysis of hearts during EVHP. Our model generates pressure-volume loops, allowing for the computation of ejection fraction, and was verified with experimental measurements taken via echocardiography.

Conclusion:

This promising tool demonstrates the unique opportunity to utilize mathematical modeling in the assessment of donor hearts, streamlining their performance evaluation. Ultimately, a more accurate assessment of donor hearts on EVHP may improve our utilization of available donor hearts, addressing the donor organ shortage that continues to limit transplant capabilities.

1. Introduction

Cardiac transplantation stands as the final treatment for end-stage heart failure. However, despite the recent increase in annual heart transplants in the United States, the waitlist continues to grow at a faster pace [1], resulting in ongoing donor-recipient mismatch and approximately 20% waitlist mortality. Surprisingly, a significant contribution seems to arise from underutilization of donor organs rather than shortage of offered organs. Previous studies have showed the potential supply of deceased donor hearts greatly outnumbers the actual number of transplanted cases [2], highlighting the urgent need for more effective management of the existing supply.

Primary graft dysfunction (PGD) is the primary cause of post-transplant morbidity and mortality [3] and affects more than 28% of transplanted hearts [4]. Given this risk, only hearts from optimal donors with exceptional function are considered for transplantation, leading to only 33% of available organs being accepted [5]. Since acceptance criteria for donor hearts remain highly subjective, many hearts are discarded due to uncertainty in their potential performance. Nevertheless, it has been shown how intensive hormonal treatment can help increase dramatically the ejection fraction and make those hearts transplantable [6]. Numerous studies consistently show that the potential supply of deceased donor organs far surpasses the actual number transplanted. A more efficient harnessing of this available supply has the potential to save thousands of lives annually in the U.S. alone and could ultimately eliminate the current waitlists [7-9].

1.1. Ex-vivo heart perfusion

Novel perfusion techniques present new opportunities to evaluate and potentially improve donor heart function prior to implantation. Historically, static cold storage has been the standard, using hypothermic temperatures (storage on ice) to slow down cellular metabolism. For example, the Paragonix SherpaPak is a cutting-edge organ preservation device designed to standardize donor heart preservation. It provides physicians with reliable cooling technology, supported by real-time data monitoring, to maintain optimal conditions throughout the preservation process. However, this technique is limited by inconsistent temperatures, ischemic damage, and inability to monitor cardiac function [10]. Ex-vivo machine perfusion is a novel method allowing for reperfusion and supply of oxygen and nutrients to the myocardium while removing toxins and metabolic waste products of the cells [11].

Clinically, there is one FDA approved device for normothermic ex-vivo heart preservation – the Organ Care System (OCS) developed by TransMedics Inc. This portable system allows for perfusion, monitoring and resuscitation. The heart is reanimated to a normothermic beating state, and perfused via retrograde blood to the aorta, provided by the pump. Assessment of the heart quality is possible visually and by means of the monitoring of circulating lactate levels, although the latter has been shown to be unreliable in recent studies, and the former is fraught by subjective interpretations [12-13]. The use of such technology has allowed a 15–20% increase in transplant volumes by using hearts that would have been otherwise rejected [14]. Nevertheless, some limitations arise imposing practical and technical challenges, including cost and unknown optimal perfusion settings [15]. Another critical disadvantage is the basic evaluation of the hearts. In the OCS, the chambers of the heart are empty and perform no work, thus making it impossible to perform functional assessments. Additionally, multiple recent studies have shown the inability of lactate trends to predict post-transplant function and the PGD risk [12,16]. Accurate and thorough heart function assessment is essential. It enables precise evaluation of donor hearts that meet current acceptance criteria and helps reduce PGD rates. Additionally, a dependable evaluation allows us to consider hearts that currently do not meet transplant criteria, and after resuscitation and potential therapeutic interventions, convert them into viable transplant options. To accomplish this, a consistent and quantifiable method of assessing heart function is necessary [17].

1.2. Computational models

Cardiac catheterization remains the gold standard for hemodynamic evaluation of cardiac function [18]. However, this invasive method is limited by accessibility and not yet applicable in an ex-vivo setting. The ability to estimate hemodynamics without catheterization would allow for the quantification of important circulatory function metrics on the dynamics of the circulatory system and cardiac function workload. Computational modeling is one such approach. Previous studies have demonstrated the success of computer-based simulations in modeling the cardiovascular system in physiologic settings, playing a major role in the diagnosis of cardiovascular diseases [19-20].

Due to the high computational cost and ethical and practical concerns with regard to collecting input data for high-dimensional patient-specific simulations, there is an extensive interest in leveraging reduced-order models, referred to as lumped parameter models (LPMs) to delineate the corresponding blood hemodynamics as a function of time [19,21]. Through a hydraulic-electrical analogy [22-23], LPMs offer a simple yet powerful means of modeling the cardiovascular system. More promisingly, their reliance on ODEs and low computational cost renders them ideal for real-time simulations readily done at the bedside [24]. By means of a fast patient-specific parameter estimation and simulation of hemodynamics, patients could be monitored in real-time with LPM [25]. Beyond offering a general understanding of cardiovascular system function, LPMs enable more ambitious applications, as they can be used in conjunction with more sophisticated approaches to yield more insightful results [26-32].

1.3. Problem and scope

Our previous efforts have developed an experimental ex-vivo perfusion system for the evaluation and optimization of donor hearts [33]. This advanced system streamlines the assessment of heart condition and rehabilitation, providing a robust platform for evaluating and improving cardiac health prior to transplantation. In this study, we developed a reliable LPM to simulate the behavior of the heart under the framework of the ex-vivo heart perfusion setup. We began by constructing a preliminary mathematical model designed to accurately replicate the hemodynamics observed in an in vivo cardiovascular system. This initial model was then validated against published in vivo data to ensure its accuracy. In subsequent stages, we adapted and refined the model to suit our specific ex vivo setup. The overarching objective was to extract cardiac and hemodynamic parameters and assess the function of the heart using mathematical modeling. The development of such a method would provide an accessible, objective tool to predict heart function prior to transplantation, ultimately improving clinical decision-making and potentially improving utilization of available donor hearts.

2. Methods

2.1. Experimental Setup

Our team has developed an in-house ex-vivo heart perfusion system capable of assessing heart function under various settings [33]. In the Langendorff mode, mimicking the clinical OCS method of perfusion, pump-generated retrograde flow through the aorta perfuses the heart and coronary arteries (Figure 1A). In this mode, the heart remains empty, and the left atrium is open. Consequently, this setup has limited capability to evaluate ejection fraction or precise hemodynamics. Under left heart loading conditions, the left atrium is cannulated, allowing the heart to fill with blood and actively pump blood forward antegrade into the aorta (Figure 1B). In this scenario, real-time, continuous measurements of intraventricular pressure can be obtained from the left ventricle, and the heart be evaluated with echocardiography for both structural and functional performance assessments. The original setup, following the schematic, is represented in Figure 1C. The circuit has similar structure for both modes, with additional connections and clamps during loading enabling flow from the pump and oxygenator to switch from filling the Windkessel to instead supply the preload reservoir with oxygenated perfusate. In both modes, the coronary sinus empties into the unloaded and open right heart and into the custom heart box, where it flows down via gravity to the lower collection reservoir.

Figure 1.

Our ex-vivo perfusion circuit for large animal hearts, including humans, in (A) unloaded and (B) loaded modes. In the unloaded (Langendorff) mode, blood flows and perfuses the heart via gravity from the compliance chamber and back to the reservoir. A pulsatile pump synchronized to heartbeat via surface EKG signal counter pulses with the heart during diastole pushing perfusate through an oxygenator into a compliance chamber. In the loaded mode, heart function is assessed using surface echo and intraventricular pressure-volume loops as heart pumps towards the compliance chamber. (C) Real setup.

In the Langendorff mode, perfusate is driven through the system by the pump located at the bottom of the setup (below the heart). As it travels to the compliance chamber at the top (above the heart), perfusate passes through an oxygenator. The compliance chamber (Windkessel bag) allows the perfusate to flow down to the heart via gravity and the built-up pressure. With a competent aortic valve, the perfusate flows into the coronary arteries, and the other cardiac chambers remain open to air and empty.

After one hour of Langendorff perfusion, loading configurations were established. In the loaded mode, the perfusate does not flow directly into the Windkessel bag after oxygenation. Instead, it is directed to another chamber, the preload chamber. This allows the perfusate to fill the left atrium under a certain pressure (8-11 mmHg) and load the left heart. The heart then pumps the perfusate out of the left ventricle against the Windkessel bag, which provides afterload. The Windkessel bag, due to its compliance, can expand or contract. After the heart ejects blood, the bag returns the perfusate to the heart to perfuse the coronaries. At this time, echocardiography images were acquired with a standard epicardial ultrasound probe. A de-aired, 50ml saline bag was positioned between the probe and myocardium to provide adequate standoff to capture the entire endocardium of each view. Image acquisition occurred at two time points: A) at the beginning (2-hour mark into perfusion) and B) at the end (4-hour mark into perfusion) of loaded graft perfusion. Four views were obtained at each timepoint: short axis parasternal/mid-papillary view, 4-chamber apical view, 2-chamber apical view, and 3-chamber apical view. Images at their respective time points were uploaded to the SYNGO platform and analyzed for ejection fraction and myocardium deformation (strain analysis) via the TomTec platform. Flow probes and pressure transducers were placed on the aorta-to-Windkessel line (Langendorff & Loaded) and the LA Preload line (Loaded only), and a thin catheter pressure probe was also placed in the LV (Loaded only). The biochemical parameters of experiments are presented in the supplementary materials.

Note that the ex vivo system and the subsequent mathematical modeling presented herein is only an approximation of the in vivo system and can only be used as an approximation tool for determining the function of the heart. In particular, the ex vivo perfusate is blood based with other crystalloids added to maintain physiologically-normal pH and nutrients. However, the perfusate is significantly more dilute than blood with hematocrit range of 10-13%.

2.2. Physiological LPM for in vivo cases

We selected LPM to achieve computational efficiency and real-time performance, which is essential for transplant decision-making where global heart metrics like pressure-volume (PV) loops are more critical than localized hemodynamic details. The LPM integrates seamlessly with our EVHP setup, enabling immediate assessment of do-nor heart performance. While 2D and 3D models provide detailed hemodynamics using finite element or computational fluid dynamics simulations, their high computational cost and lack of real-time feasibility limit their practicality for whole-system cardiovascular modeling. Our choice of the LPM balances accuracy and real-world application, making it an optimal tool for timely and effective transplant evaluation.

A preliminary mathematical model was developed to accurately replicate the hemodynamics observed in an in-vivo cardiovascular system, with the intention of adapting it for a specific ex-vivo setup in subsequent stages. First, we explain the theoretical background and modeling equations for each individual component of the cardiovascular system. Then, we detail how these equations were adapted to replicate our ex-vivo experiment.

2.2.1. Heart chambers

2.2.1.1. Left ventricle

The contractile state of the left ventricle (LV) is commonly characterized using the concept of time-varying elastance (E(t)) [34]. This well-established approach describes the relation between chamber pressure and volume throughout the cardiac cycle (Eq. 1):

$$E(t)=\frac{P(t)}{V(t)-V_0}$$ (Eq. 1)

where: P(t) and V(t) are the chamber pressure and volume, respectively and $V_0$ is the initial volume. The shape of the curve has been studied, leading to the development of various elastance functions, such as those based on summation of Gaussian functions as well as Boltzmann distributions. Nevertheless, the double-Hill function has been used for the purpose of this model due to its physiological resemblance to pressure, flow, and volume waveforms (Eq. 2) [35]:

$$E(t)=\alpha ({\mathrm{E}}_{\max }-E_{min})\left(\frac{{\left(\frac{t}{{\tau }_1}\right)}^{m1}}{1+{\left(\frac{t}{{\tau }_1}\right)}^{m1}}\right)\left(\frac{1}{1+{\left(\frac{t}{{\tau }_2}\right)}^{m2}}\right)+E_{min}$$ (Eq. 2)

Here, ${\mathrm{E}}_{\max }$ and ${\mathrm{E}}_{\text{min}}$ represent maximal and minimal elastance (i.e., stiffness of the chamber). For the left ventricle, ${\mathrm{E}}_{\max }$ corresponds to the end-systolic elastance, and ${\mathrm{E}}_{\text{min}}$ characterizes the passive diastolic elastance of the chamber. Dimensionless parameters ${\tau }_1$, ${\tau }_2$, ${\mathrm{m}}_1$, and ${\mathrm{m}}_2$ define the shape of the elastance curve. Parameters ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ represent the rate of chamber contraction and relaxation (i.e., slope of the curve), respectively. Parameters ${\tau }_1$ and ${\tau }_2$ define the length of the two stages in the cardiac cycle, systole and diastole, respectively. ${\tau }_1={\alpha }_{\mathrm{s}}\cdot \mathrm{T}$ and ${\tau }_2={\alpha }_{\mathrm{d}}\cdot \mathrm{T}$ thereby determining the length of systole and diastole relative to each cardiac cycle (T). Finally, the scaling factor α ensures a maximal elastance of $\alpha$ and minimal of ${\mathrm{E}}_{\max }$.

2.2.1.2. Left atrium

Coupling of the volume and pressure in the left atrium (LA) chamber was performed using the same procedure defined in Eqs. 1-2. To account for the phase lag between the onset of contraction in the two chambers, LA contraction was initiated at 0.85 of the cardiac cycle, with the initial point of LV contraction serving as the reference.

The time varying elastance is often represented as a variable capacitor component, enabling the generation of an analog periodic signal that simulates the contraction and relaxation of the heart throughout the cardiac cycle, and ultimately making the chamber act like a voltage generator driving the system.

2.2.2. Heart valves

The net pressure gradient formulation was used to model the unidirectional flow of blood through the valves during the cardiac cycle, taking into account factors such as flow rate (including acceleration and deceleration) and energy losses (Eq. 3).

$$\Delta P_v(t)=R_v{Q}_v(t)+L_v\frac{d{Q}_v(t)}{dt}$$ (Eq. 3)

In this equation, ${\mathrm{Q}}_{\mathrm{v}}(\mathrm{t})$ represents the instantaneous flow rate across the given valve, while ${\mathrm{R}}_{\mathrm{v}}$ and ${\mathrm{L}}_{\mathrm{v}}$ denote the valve resistance and inertance, respectively. The first term characterizes pressure losses, whereas the second term accounts for the acceleration and deceleration of the flow. This representation conceptualizes the valves as a combination of a diode, representing their one-way functionality, in series with a resistor and inductance. The transition from a closed to an open state, mimicked by a diode, is triggered by the pressure gradient across the valve. This mechanism enables unidirectional flow of blood through the system, reflecting the physiological behavior of the heart valves.

2.2.2.1. Aortic Valve

Adapting the model to encompass the behavior of the aortic valve was imperative to accurately account for its distinct characteristics during LV ejection. The general equation remains applicable, with the substitution of the appropriate resistance and inductance parameters specific to the aortic valve:

$$\Delta P_{AV}=\frac{\rho }{2E_{L}C{o}_{\mid A{V}^2}}{Q}_{AV}^2(t)+\frac{2\pi \rho }{\sqrt{E_L{Co\mid }_{AV}}}\frac{\partial Q_{AV}(t)}{\partial t}$$ (Eq. 4)

$$E_L{Co\mid }_{AV}=\frac{({EOA\mid }_{AV})A_{AO}}{A-{EOA\mid }_{AV}}$$ (Eq. 5)

In this equation, $\Delta P_{AV}$ is the net pressure gradient and ${\mathrm{Q}}_{\text{AV}}(\mathrm{t})$ represents the instantaneous flow rate across the aortic valve and $\rho$ is the blood density. ${EOA\mid }_{AV}$ and ${\mathrm{A}}_{\text{AO}}$ represent the effective orifice area of the aortic valve and the cross-sectional area of the ascending aorta, respectively, and $A$ is the cross-sectional area of the valve annulus. The energy loss coefficient ($E_L{Co\mid }_{AV}$) is included to accommodate the pressure recovery phenomenon commonly observed in patients with aortic stenosis [36]. This way, $\frac{2\pi \rho }{\sqrt{E_L{Co\mid }_{AV}}}$ corresponds to the constant inertance of the valve model, while $\frac{\rho }{2E_L{Co\mid }_{A{V}^2}}{Q}_{AV}^2$ signifies a variable resistance where $\rho$ denotes the blood density.

2.2.2.2. Mitral valve

The identical analytical formulation, based on Eq. 4, is employed to characterize the instantaneous net pressure gradient across the mitral valve during LA ejection. However, no pressure recovery phenomenon is considered given the large LV volume, which renders the energy loss coefficient irrelevant in this context [30]:

$$\Delta P_{MV}=\frac{\rho }{{2EOA\mid }_{MV}^2}{Q}_{MV}^2(t)+\frac{M_{MV}}{{EOA\mid }_{MV}}\frac{\partial Q_{MV}(t)}{\partial t}$$ (Eq. 6)

In this equation, $\Delta P_{MV}$ is the net pressure gradient and ${\mathrm{Q}}_{\text{MV}}(\mathrm{t})$ represents the instantaneous flow rate across the mitral valve and $\rho$ is the blood density. ${EOA\mid }_{MV}$ and ${\mathrm{M}}_{\text{MV}}$ represent the effective orifice area and inertance, respectively.

2.2.2.3. Regurgitation

Various valvular diseases, such as aortic stenosis or mitral regurgitation, can disrupt the normal opening or closing of valve leaflets. By extending the previous formulation, a second diode was introduced for each valve, allowing for bidirectional flow.

2.2.3. Systemic and pulmonary segments

The remaining modules pertain to vasculature segments, encompassing both systemic and pulmonary systems. Each component was modeled by a three-element Windkessel model, comprising a resistor to account for frictional losses, inductor to represent mass inertia, and a capacitor to capture elasticity of the vessel walls. In the modeling of the pulmonary venous system, a constant pressure was incorporated to signify perfusion pressure in the capillaries, along with segments representing the pulmonary capillaries and veins [31]. Moreover, a pulmonary flow waveform was simulated to further characterize the dynamics of pulmonary circulation:

$$Q_{pv}(t)=Q_{mpv}\sin \left(\frac{\pi t}{t_{ee}}\right),t\le t_{ee}$$ (Eq. 7)

$$Q_{pv}(t)=0,t_{ee}<t\le T$$ (Eq. 8)

Herein, ${\mathrm{Q}}_{\text{mpv}}$, ${\mathrm{t}}_{\text{ee}}$, and $T$ represent the mean flow rate of the pulmonary valve, the end-ejection time, and the cardiac cycle period, respectively. This formulation introduces the concept of returning flow towards the left heart. In our scenario, this is of great importance because, in the actual ex-vivo setup, it is the left heart that is being perfused, as will be further elaborated in the subsequent section.

2.2.4. Coronaries

Given the significant role of the coronary arteries, particularly during perfusion in the ex-vivo setup, their blood flow has been meticulously modeled. The analytical formulation for coronary arteries differs from other vascular segments due to their unique characteristics. During myocardial contraction, blood flow in coronary arteries momentarily reverses as a result of the contractile forces exerted on them. Consequently, maximum blood flow in the coronary arteries occurs during early diastole, after the vessels are relaxed and not under compression, contrary to most blood vessels where flow peaks during systole [19]. For our purposes, a simplified model was employed, including the three major branches of the coronary arteries. A pressure source was introduced to reflect the ventricular pressure exerted on the coronary vasculature. The remainder of the coronary network was modeled based on principles analogous to those used for other vessels. Overall, the model includes coronary arterial resistance and compliance, coronary arterial microcirculation resistance, myocardial compliance with sustained pressure, and finally coronary venous microcirculation resistance and coronary venous resistance [37].

2.2.5. Patient-specific parameter estimation

Parameter values were typically derived from literature data, as invasive measurement for each case would contradict the foundational purpose of leveraging the LPM. However, to render the model case-specific, additional inputs must be gathered non-invasively. These inputs can either be directly used or integrated into a posterior parameter framework to tailor the model to individual patients. The key inputs necessary for this customization include:

• Flow inputs: Flow data were continuously measured during tests. This data can serve as an input in the LPM, facilitating the optimization of certain model parameters, particularly those associated with the flow returning to the heart (${\mathrm{Q}}_{\text{pv}}$). By adjusting these parameters, the model can be fine-tuned to produce a flow output akin to the measured values.

• Time inputs: Using ECG and/or Doppler echocardiography, the duration of cardiac cycle ($T$) and time of ejection (${\mathrm{t}}_{\text{ee}}$), crucial characteristics of the model, can be accurately determined.

• Valve/regurgitation inputs: The area of the respective valve/regurgitation ($A$) and the effective orifice area ($EOA$) can be derived by tracing the pulse wave flow envelope using Doppler.

• Volume inputs: End systolic (ESV) and end diastolic volume (EDV) were also measured using surface echocardiography, being used to adjust the ranges of starting and ending volumes (end-systolic and end-diastolic LV volumes) in the PV loop.

• Chamber inputs: After calculating the cardiac cycle duration ($T$), the elastance function characteristic of each chamber can be determined using (Eq. 2).

• Parameter estimation: Depending on the directly measurable inputs, additional parameters may require adjustment through a parameter estimation approach. The selection of such parameters and the optimization process will be discussed later.

Definition of input parameters and their identification were essential for simulating the model. Due to the unavailability of in-vivo porcine heart measurements, various resistances, capacitances, inductances, and other input parameters were estimated based on literature values for our preliminary, qualitative assessment aimed to understand the behavior of the developed model [31]. Clinically measurable inputs, obtained through Doppler echocardiography, such as flow inputs, time and chamber inputs (cardiac cycle, shape-related elastance parameters), and valve parameters (effective orifice area) were derived from the literature [30-32].

The model underwent numerical analysis within the MATLAB Simscape (MathWorks, Inc.) environment. The initial model, constructed using the described mathematical formulations, comprised the following modules: pulmonary return flow, including the pulmonary arteries, veins and capillaries; the left atrium chamber, the mitral valve and its regurgitation; the left ventricle, the aortic valve and its regurgitation; the aorta, and finally the systemic arterial and venous tree. Additionally, the coronary arteries were incorporated into the model to provide a comprehensive representation of cardiovascular dynamics.

We did a sensitivity analysis to determine the influence of various parameters on the model’s output, and identified and eliminated those with low impact. To achieve case specificity, a multi-parameter sensitivity analysis was performed using Monte Carlo simulations, generating random values for different parameters. The influence of each parameter on the model was assessed by comparing the resulting output signals to those of an initial case.

2.3. Patient-specific framework and validation

Benchmarking was performed by comparing our current LPM against other experiments and published models [30-31]. We aimed to verify if our model could reproduce similar results, thus signifying its flexibility in capturing various cases and validating its accuracy and reliability.

To accurately simulate a case-specific heart, certain model parameters required optimization to ensure that the LPM could reproduce physiological waveforms. Following the sensitivity analysis, parameters associated with pulmonary vasculature were excluded from optimization and kept constant. Conversely, heart-related and systemic circulation parameters were included for adjustment.

Nonlinear optimization was employed to estimate the parameters by minimizing the sum of squared errors (SSE) between the waveforms generated by our model and those obtained from specific cases in published models and clinical measurements. Specifically, the model was validated against both healthy and non-healthy cases. For each measurement, the cost function was defined as:

$$J(\overline{x},x)={\sum }_{i=1}^{N_i}{\left(\overline{x}(i,\theta )-x(i)\right)}^2$$ (Eq. 9)

where $\overline{x}(i,\theta )$ corresponds to the simulation output of the model at time step $i$ with parameters $\theta$, and $x(i)$ representing the measured values at that time step. The objective of the optimization was to minimize the sum of the individual measurements type deviation defined as cost functions:

$$C(\theta )={\sum }_{i=1}^{d}J({\overline{x}}_i(\theta ),x_i)$$ (Eq. 10)

where $i=1,2,\dots ,d$ represents each of the measurement types we are comparing our model to, such as LV pressure, LV volume, or aortic pressure.

The Simulink Design Optimization toolbox was used to optimize the LPM response using the trust region reflective algorithm implemented in MATLAB, which has shown efficiency in parameter estimation framework for LPMs [38]. Furthermore, within the patient-specific parameter estimation framework, we defined parameter intervals to ensure solutions remained within physiological ranges. Based on initial parameter values in [31], we set intervals as follows: [${\mathrm{p}}_0∕2$, $2{\mathrm{p}}_0$] for resistance components, and [${\mathrm{p}}_0∕5$, $5{\mathrm{p}}_0$] and [${\mathrm{p}}_0∕6$, $6{\mathrm{p}}_0$] for inertance and compliance components, respectively. Parameters characterizing the heart chambers were restricted to [${\mathrm{p}}_0∕2$, $2{\mathrm{p}}_0$].

It is important to note that the various measured waveforms used in the optimization exhibit considerable variation in magnitude. Volumes, flow rates, and pressures all have distinct ranges of values. To address this, both outputs and measurement data were scaled prior to their use in the cost function calculation, thereby accommodating these differences. Scaling was typically performed out by the Min-Max approach.

2.4. LPM of the ex vivo setup

To adapt the LPM to the ex vivo setup, we retained the heart-related elements (chambers, valves, coronaries) from the previous in vivo LPM, while new components representing the tubing and other setup elements were incorporated into the model (Figure 3). The majority of these elements were included in the model as constant parameters since they remained equivalent between runs.

Figure 3.

LPM for the ex-vivo perfusion setup.

The tube connections in the loaded mode were represented by different three-element Windkessel models. Compliance chambers, such as the Windkessel bag, were only modeled with a capacitance. Additionally, the pump driving the setup was included in the model.

2.5. Swine Ex vivo heart perfusion model

The ex-vivo perfusion model was tested using swine hearts, procured and placed on our in-house EVHP set-up. The study was approved by IACUC (protocol #2023N000042, approved March 24, 2023) and is in concordance with animal regulatory requirements of Massachusetts General Hospital. Yorkshire swine hearts were procured by a transplant surgeon after circulatory death, and reanimated on EVHP. Oxygenated Krebs-Henseleit solution with 7-10% hematocrit was used as circulating perfusate. The hearts were maintained on EVHP for 4 hours, with 1 hour of Langendorff perfusion, followed by 3 hours under loading conditions.

During perfusion, left ventricle pressure, aortic tube pressure, aortic flow rate, and flow entering the left atrium were measured and utilized to optimize our model. The parameter estimation framework employed remained consistent, involving the minimization of the difference between model output waveforms and those measured in the setup, evaluated through the sum squared error (SSE). In this process, parameters representing setup components were maintained as constants, while heart-related parameters were adjusted to align with the observed behavior.

Using the pressures and flow waveforms generated by the model, heart function was assessed through a performance analysis. Proposed metrics for this evaluation included the PV loop, from which parameters such as stroke volume and ejection fraction by the left ventricle could be derived. These metrics were compared to values obtained through surface echocardiography conducted at specific timestamps during the runs.

2.6. Model validation

To benchmark the capabilities of the proposed model, two case studies were examined: a healthy and a pathogenic case (Figure 4). Data for these cases were sourced from literature, which provided the outputs generated from clinical non-invasive measurements [30-31]. The waveform data served as the basis for optimizing our model parameters to align the LPM results with their observed behavior. Digitalization tools were employed to extract the corresponding data points from the graphs, facilitating their integration into our patient-specific parameter estimation framework.

Figure 4.

Model output during one cardiac cycle after optimization compared to (A) healthy case and (B) a non-healthy case for validation.

The results after parameter estimation for the healthy subject demonstrate the realistic reproduction of various hemodynamic variables for which it was optimized. These variables include left ventricular pressure, aortic pressure, aortic flow, mitral flow, left atrial volume and left ventricular volume. The agreement between the model outputs and observed data is notable, qualitatively in terms of shape, amplitude, and timing. Systolic and diastolic pressures for both the left ventricle and aorta show very close agreement, with a root mean squared error (RMS) of only 2.27 mmHg and 1.29 mmHg for LV and aorta pressure, respectively. Flow variables also demonstrate excellent correspondence, with an RMS error of 5.9 mL/s for the aortic flow. In terms of LV volume, the RMS error is 3.6 mL, indicating good agreement with the observed stroke volume ejected. Importantly, there is a strong matching between the waveforms not only in amplitude, but also in time intervals. This precise alignment is crucial for accurately capturing the dynamics of the cardiac cycle, where the mitral inflow defines the filling time of the left ventricle, while aortic outflow determines its ejection time, as well as the timings between filling and ejection for the isovolumetric contraction and between ejection and refilling for isovolumetric relaxation.

In the non-healthy case representing aortic stenosis, the patient exhibited severe aortic stenosis (EOA = 0.55 cm²), mild aortic regurgitation (AR), and mild mitral regurgitation (MR), accompanied by a decreased ejection fraction of 60–65% and stroke volume of 52 mL. Once again, the model accurately captures the shapes, ranges, and timings of various waveforms. Calculations reveal an RMS error of 5.2 mmHg and 3.1 mmHg for LV and aorta pressures, respectively. Notably, the model accurately replicates the dynamics of the so-called dicrotic notch in the aorta, a physiological effect occurring at the closing of the aortic valve. This phenomenon involves a brief increase in arterial blood pressure due to the surge of blood against the valve cusps as the aortic valve closes. Flow variables also demonstrate strong correspondence, with RMS error of 3.5 mL/s for the aortic flow. Regarding LV volume, the RMS error is 1.7 mL, indicating excellent agreement with the reported stroke volume and ejection fraction. Though several parameter values were not provided for the model simulation, the reported value of 0.55 for ${\text{EOA}}_{\text{av}}$, a critical indicator of the stenosis severity, was matched after optimization in our model, further validating its accuracy.

3. Results

The model results after parameter estimation for the swine heart experiments are shown in Figure 5. Depending on the case, heart parameters are re-estimated individually, while setup-related parameters remain constant across all cases. Various hemodynamic variables from the model show favorable qualitative agreement compared to those measured in the setup. LV pressure demonstrates notable correspondence in terms of systolic and diastolic pressure, yielding an RMS error of 3.31 mmHg. The pressure at the aortic tube accurately captures the characteristic two-peak waveform, resulting in an RMS error of 3.48 mmHg. Moreover, the flow through the aortic tube effectively represents its bidirectional flow, pumping out of the heart and back to it, with calculations revealing an RMS of only 1.33 mL/s. Finally, the flow filling the left atrium from the prechamber shows good correspondence. Despite exhibiting some periodic behavior, the flow remains relatively constant due to the prechamber acting as a constant pressure source, resulting in an RMS error of 0.23 mL/s.

Figure 5.

Model output during one cardiac cycle following optimization (A) compared to the measured swine heart data obtained in the EVHP. Performance analysis (B) exctracting aortic flow and PV loop.

Aortic flow pumped by the heart in each cardiac cycle was extracted, ultimately determining its stroke volume. In plotting the PV loop, the LV pressure was graphed alongside the volume of the chamber. While the volume range effectively represents the stroke volume passing through the aortic valve, initial and endpoint volumes within the interval, crucial for ejection fraction calculation, are nonsensical due to transient effects in the model's initial cycles, resulting in negative volumes. Therefore, proper adjustment of the volume interval of the PV loop to align with the system’s operation was imperative. To this end, using echocardiography data, the end-diastolic (or end-systolic) volumes were determined. By employing one of these volumes alongside the stroke volume, the remainder could be computed. Subsequently, the corresponding PV loop and ejection fraction (EF = SV/EDV) were evaluated. The swine heart exhibited an estimated LV ejection fraction of 32% showing great agreement with experimentally measured data obtained via echocardiography (31%).

4. Discussion

Given the persistent shortage of hearts for transplantation, it becomes imperative to employ evidence-based heart evaluation methods to accurately quantify case-specific heart function. In response to this need, we previously developed a novel in-house ex-vivo heart perfusion system capable of assessing heart function [33]. A lumped parameter model has also been developed to streamline performance analysis and evaluation by integrating both the heart and the perfusion setup. Furthermore, the mathematical model offers the advantage of extracting information from different locations, eliminating the necessity for invasive measurements in real-life scenarios.

An initial preliminary mathematical model was devised to first capture the behavior of an in-vivo heart. Comprehensive parametric analyses and sensitivity studies were performed to discern the contribution of various parameters to the model outcomes and to identify dominant parameters for calibration. While certain parameters were derived from literature values, others required fine-tuning. The model was rigorously validated against experimental data and published models, successfully replicating both healthy and aortic stenosis cases.

The in-vivo mathematical model was refined to accommodate the in-house ex-vivo perfusion setup, specifically focusing on performance analysis of tested hearts under the loaded mode of the system. While retaining heart-related components, adjustments were made to incorporate segments of the setup, such as tubing and compliance chambers. Through experimental characterization of these segments, their properties were established and integrated into the model as constant parameters regardless of the heart being perfused. Meanwhile, heart-related parameters were optimized to ensure accurate representation. Subsequent experiments, particularly focusing on a swine heart case, allowed for thorough performance analysis. Leveraging outputs from these experiments, alongside echocardiography measurements obtained during runs, crucial heart function metrics were extracted. Notably, the computation of PV loops facilitated the determination of stroke volume and ejection fraction, demonstrating strong alignment between our model and experimental echocardiography measurements.

LPMs serve as valuable tools to explore the heart performance and its effects on both local and global hemodynamics. In fact, such models have been leveraged to investigate the performance of clinically approved medical devices [19]. Simulating the heart behavior offers a significant advantage as it allows for the extraction of quantitative information about the cardiac function. Pressure-volume loops provide a critical understanding of cardiac metrics by correlating chamber pressure with volume. Metrics like ejection fraction of the ventricle, contractility, and stroke volume play a key role in evaluating heart health. Moreover, this simulation method provides valuable insight into the impact of different cardiovascular diseases, aiding in diagnosis. Since obtaining data for PV loops typically involves invasive procedures, personalized lumped parameters become exceedingly valuable in this context.

Our model's reliance on personalized parameters is not a limitation but a significant translational strength, particularly in cardiac transplantation where precision and adaptability are critical. Personalized parameters allow the model to account for variations in donor heart function, graft quality, and recipient requirements. This precision enables clinicians to make informed decisions, such as determining whether a donor heart can generate the necessary aortic pressure at a given left atrial pressure to meet a recipient’s needs. If a mismatch is identified, the model facilitates selecting an alternate, more suitable recipient, improving graft utilization and clinical outcomes. By integrating patient-specific data, the model aligns with the principles of precision medicine, offering tailored predictions that go beyond the generalized thresholds of traditional methods. Its ability to adapt to diverse clinical scenarios, including complex cases of organ matching, ensures that patient specificity enhances its utility. This adaptability, combined with its focus on individualization, makes the model an invaluable tool for improving decision-making in transplantation and broader clinical contexts.

Although our study is very promising, several limitations warrant consideration. Moving forward, key steps can be taken to enhance the impact and usability of the developed tool. Firstly, the proposed model relies on elements of the perfusion set-up at our institution and may not be broadly applied to in-house perfusion systems at other centers without adjusting the model components. Second, our model focuses on LV loading. While the biventricular ex vivo perfusion system offers the most physiologically accurate representation of cardiac function, it presents significant complexities in implementation and operation. Given that the ischemia-reperfusion insult impacts both ventricles similarly, we propose that a single-ventricle loading model, particularly focused on the LV, is sufficient for assessing cardiac performance. The LV plays a primary role in systemic circulation and is typically more affected during ischemic events, making it an appropriate focus for functional evaluation. This streamlined approach, with the potential to expand the analysis to biventricular loading, simplifies the setup and reduces experimental variability while still providing reliable insights into cardiac function and recovery. Moreover, the creation of a user-friendly graphical interface accessible at the bedside of the perfusion setup is crucial. This interface should offer comprehensive heart performance analysis in real-time, allowing for immediate insights into the heart's function. Additionally, continual refinement of the model is essential to adapt to any changes in the setup. The optimization process should be streamlined to ensure swift updates, enabling real-time usage of the tool alongside perfusion activities. Moreover, while not addressed in the current project, research into the identifiability of model parameters is warranted due to their clinical significance. Understanding which parameters are identifiable can improve the accuracy and reliability of the model's predictions. However, it is important to note that even with non-identifiable parameters, the model can still yield valuable insights and provide useful predictions. Lastly, further analysis to evaluate the model’s ability and robustness in assessing heart function across various perfusion runs should be conducted. This will help validate the model's effectiveness in different scenarios and ensure its reliability in clinical settings. By pursuing these further steps, the developed tool can continue to evolve and make significant contributions to the field of heart perfusion research and clinical practice.

5. Conclusion

Our findings underscore the immense potential of our model to effectively assess heart performance within the context of the ex-vivo perfusion setup. Such a tool holds significant promise in enhancing our ability to quantify the hemodynamic properties of individual hearts through minimally invasive monitoring. Ultimately, our model addresses the unmet need for quantitative evaluation of heart function prior to transplantation.

Figure 2.

Workflow of the patient-specific parameter estimation framework.