Proposed method
The TVE curve is defined:
$$e(t) = \frac{{P_{lv} \left( t \right)}}{{V_{lv} \left( t \right)  V_{d} }}$$
(1)
where P
_{
lv
} is the pressure in the left ventricle, V
_{
lv
} is the volume in the left ventricle and V
_{
d
} is the ‘dead space’ volume in the ventricle [6, 23]. Thus, the TVE curve is defined by two waveforms (P
_{
lv
} and V
_{
lv
}) and one constant (V
_{
d
}). These values can be measured directly, but doing so is not clinically feasible [24].
The proposed method approximates these waveforms (V
_{
lv
}, P
_{
lv
}) and constant (V
_{
d
}) using three inputs, as shown in Fig. 1. These inputs include continuously sampled aortic pressure waveforms (P
_{
ao
}) and heart rate (HR), data which is typically available in a modern ICU. The final input required is baseline endsystolic (V
_{
es
}) and enddiastolic (V
_{
ed
}) Volume, obtained from a brief echocardiography reading, which is increasingly available in a clinical setting [25].
The overall goal of this method is to use the clinically available inputs P
_{
ao
}, HR and baseline V
_{
es
} and V
_{
ed
} to determine the outputs P
_{
lv
}, V
_{
d
} and V
_{
lv
}, and thus the TVE curve e(t) as set out in Fig. 1 and Eq. 1. This resulting TVE curve can be experimentally validated against the TVE curves derived from experimentally measured P
_{
lv
} and V
_{
lv
} in animal models. The availability of a nearby continuous pressure measurement (P
_{
ao
}) forms an effective basis for the continuous approximation of P
_{
lv
}, and the availability of a baseline volume measurement (V
_{
es
}) forms an effective basis for the approximation of baseline V
_{
d
}. However, there is no continuous volume measurement available for continuous approximation of Vlv, resulting in this process being considerably more involved. Thus, the shaded central Region III in Fig. 1 is considerably more complicated than Regions I and II.
While the overall method involves approximating two output waveforms (V
_{
lv
}, P
_{
lv
}) from a single input waveform (P
_{
ao
}), it’s important to note that all three waveforms (P
_{
lv
}, V
_{
lv
}, P
_{
ao
}) have distinct features. Different regions of behaviour are governed by different physiological phenomenon, and these waveforms have been extensively characterised [26]. As such, all three waveforms are heavily interconnected and information rich, making this task more reasonable than it might first appear.
Determining P
_{
lv
} from P
_{
ao
} (Region I, Fig. 1)
The left ventricle is situated directly upstream from the aorta, separated by the aortic valve. This valve is open during systole and closed during diastole. As such, if aortic valve resistance is neglected, P
_{
lv
} is equivalent to P
_{
ao
}, with a slight phase lag (δ) during the majority of systole (Section P. 1, Fig. 2). While aortic valve resistance is nonnegligible in conditions such as aortic stenosis [26], valve dysfunction of any type is present in only 3.61% of CVD mortalities in the US [27]. Further, such conditions are typically chronic in nature, relatively easily diagnosed [28] and evolve slowly, while the method is designed to monitor short term changes in an ICU environment. As such, this assumption should not significantly affect the methods’ applicability to the vast majority of the target cohort.
During diastole the aortic valve is closed and little information is available from the aortic waveform about ventricular behaviour. However, the ventricle behaves in a largely passive manner in this region, meaning the TVE curve is typically near zero during diastole [26]. As such, a generic function consisting of two exponentials was used to approximate left ventricular pressure during diastole. In early diastole (Section P. 2, Fig. 2) an exponential decay to a fixed baseline pressure captures ventricular relaxation. In late diastoleearly systole (Section P. 3, Fig. 2), an exponential increase captures the beginning of ventricular contraction [26].
While atrial contraction contributes significantly to late diastolic filling, ventricular elastance remains largely passive in late diastole [26]. Thus the TVE curve is typically at its baseline value until the beginning of ventricular contraction in early systole. While atrial function affects the magnitude of the driver function, which is normalised in this work, and may affect its shape, this effect is indirect as the driver function represents the impact of contraction in driving pulsatile blood from the heart to the arterial system. Further, with the increasing unpopularity of pulmonary artery catheters [29] none of the typically available instrumentation in an ICU provides a clear picture of atrial behaviour. As such, while the exponential in section P. 3 is broadly intended to capture ventricular filling, no specific atrial behaviour component is integrated into this model.
Using Fig. 2, the left ventricular pressure for the n
_{
th
} heartbeat is thus defined using P
_{
ao
}:
$$t_{1} = t\left( {\frac{{dP_{ao} }}{dt}_{max} } \right)_{n}$$
(2a)
$$t_{2} = t\left( {\frac{{dP_{ao} }}{dt}_{min } } \right)_{n}$$
(2b)
$$t_{4} = t\left( {\frac{{dP_{ao} }}{dt}_{max } } \right)_{n + 1}$$
(2c)
$$t_{3} = 0.62t_{2} + 0.38t_{4}$$
(2d)
$$P_{lv} (t) = \left\{ {\begin{array}{*{20}l} {P_{ao} \left( {t_{1} + \delta < t < t_{2} + \delta } \right)} \hfill & {t_{1} < t < t_{2} } \hfill \\ {6 + \left( {P_{ao} \left( {t_{2} } \right)  6} \right)e^{{  17.5\left( {t  t_{2} } \right)}} } \hfill & {t_{2} < t < t_{3} } \hfill \\ {P_{lv} \left( {t_{3} } \right) + \left( {P_{ao} \left( {t_{4} } \right)  P_{lv} \left( {t_{3} } \right)} \right)e^{{37.5\left( {t  t_{4} } \right)}} } \hfill & {t_{3} < t < t_{4} } \hfill \\ \end{array} } \right\}$$
(3)
where: \(\delta = 0.008{\text{s}}\)
Determining V
_{
d
} from baseline V
_{
es
} (Region II, Fig. 1)
A recent method has been developed to approximate V
_{
d
} from ventricular volume measurements [30]. This approach relies on linear regression of the FrankStarling curve (SV–V
_{
ed
}) and its endsystolic equivalent (SV–V
_{
es
}) to the point where SV = 0 and ‘the ventricle cannot develop any systolic pressure’, the definition of V
_{
d
} [23]. This work also showed V
_{
d
} for a baseline, healthy pig be an approximately fixed percentage of V
_{
es
} [30]. Defining V
_{
d
} as a percentage of baseline V
_{
es
} allows approximation of baseline V
_{
d
} during the initial echocardiographic reading where measured V
_{
es
} is available. While V
_{
d
} has been shown to change with condition, there is no practical means of capturing this change short of additional echocardiography measurements. Thus, while intermittent measures are feasible, in this study V
_{
d
} is fixed at a baseline value.
$$V_{d} = 0.48 \times V_{es}$$
(4)
Determining V
_{
lv
}
from P
_{
ao
}
, HR and V
_{
d
} (Region III, Fig. 1)
Unlike pressure, little volume or flow information is readily available from the typical, clinically available instrumentation. Simulating V
_{
lv
} is thus more challenging. The shape of the ventricular volume waveform was approximated by a piecewise sine wave consisting of two sections: systole (Section V. 1, Fig. 3) and diastole (Section V. 2, Fig. 3), with a 90° phase shift at the beginning of systole. The underlying physiological behaviour might be better represented by a series of exponentials [26]. However, using sine waves achieves a similar result with considerably fewer variables involved.
Thus, six points per heartbeat (t
_{1}, t
_{2}, t
_{3} and (V
_{
ed
})_{
n
}, (V
_{
es
})_{
n
}, (V
_{
ed
})_{
n+1
}) are required to define the ventricular volume waveform. The timing associated with systole start (t
_{1}), systole end (t
_{2}), and diastole end (t
_{3}) are readily determined from the aortic pressure waveform (Fig. 3):
$$t_{1} = t\left( {P_{ao_{min}} } \right)_{n}$$
(5a)
$$t_{2} = t\left( {P_{DN} } \right)_{n}$$
(5b)
$$t_{3} = t\left( {P_{ao_{min}} } \right)_{n + 1}$$
Using existing work [31] SV can be approximated beattobeat using the aortic waveform. Thus, only one of V
_{
es
} or V
_{
ed
} is required, as SV can be used to convert between the two. The ESPVR allows for determination of V
_{
es
}, and is defined [23]:
$$P_{es} = E_{es} \times \left( {V_{es}  V_{0} } \right)$$
(6)
where E
_{
es
} is the endsystolic elastance and V
_{0} is the ventricular volume at zero pressure. Equation 6 can be rewritten:
$$P_{DN} = E_{es} \times \left( {V_{es}  V_{d} } \right)$$
(7)
where this change is justified by:

The pressures in the ventricle and aorta are roughly equivalent until the aortic valve closes, thus P
_{
DN
} is close to P
_{
es
}

V
_{
d
} and V
_{0} have similar, but distinct, physiological significance and values. The two are often used interchangeably [17, 18, 23]
Finally, it is necessary to account for E
_{
es
}, which changes in response to a number of factors including contractility [6], and loading conditions [32, 33]. Thus, Eq. 7 is modified:
$$P_{DN} = \left( {E_{c} \times HR^{3} } \right) \times \left( {V_{es}  V_{d} } \right)$$
(8)
here E
_{
es
} is defined as a function of HR and a coefficient (E
_{
C
}), with a cubic selected as it provides the best compromise between simplicity and effective tracking for the data set presented here. In particular, the cardiovascular system responds to most changes in conditions in a number of ways, including changes in heart rate and elastance. As heart rate is easily measured, it provides an easy to obtain, if incomplete, indication of cardiovascular system response, which can be used to inform an approximated elastance [34]. Further supporting evidence is provided in the validation and discussion of results.
During the echocardiography calibration, measurements for P
_{
DN
}, HR and V
_{
es
} are available [35]. Thus, using Eq. 8, a constant value for E
_{
C
} can be defined, allowing approximation of E
_{
es
} and thus determination of V
_{
es
} on a beatbybeat basis. The beattobeat ventricular volume can thus be determined:
$$V_{lv} (t) = \left\{ {\begin{array}{*{20}l} {\left( {V_{ed} } \right)_{n} + \left( {\left( {V_{es} } \right)_{n}  \left( {V_{ed} } \right)_{n} } \right)\sin \left( {\frac{{\pi \left( {t  t_{1} } \right)}}{{2\left( {t_{2}  t_{1} } \right)}}} \right)} \hfill & {t_{1} < t < t_{2} } \hfill \\ {\left( {V_{es} } \right)_{n}  \left( {\left( {V_{ed} } \right)_{n + 1}  \left( {V_{es} } \right)_{n} } \right)\left( {\frac{1}{2}\cos \left( {\frac{{\pi \left( {t  t_{2} } \right)}}{{\left( {t_{3}  t_{2} } \right)}}} \right)  \frac{1}{2}} \right)} \hfill & {t_{2} < t < t_{3} } \hfill \\ \end{array} } \right\}$$
(9)
where:
$$V_{es} = \frac{{P_{DN} }}{{\left( {E_{c} \times HR^{3} } \right)}} + V_{d}$$
(10)
$$V_{ed} = V_{es} + SV$$
(11)
Summary of proposed method
The overall derivation of the TVE curve can be summarised:
Initially or intermittently

1.
Calculate V
_{
d
} using Eq. 4 and baseline V
_{
es
} (Region I, Fig. 1)

2.
Calculate E
_{
C
} using Eq. 8, P
_{
DN
}, HR and baseline V
_{
es
} (Region III, Fig. 1)
Every heartbeat

1.
Simulate P
_{
lv
} using Eq. 2, Eq. 3 and P
_{
ao
} (Region II, Fig. 1)

2.
Determine V
_{
es
} using Eq. 10, P
_{
DN
}, HR and E
_{
C
} (Region III, Fig. 1)

3.
Determine SV using [31] and P
_{
ao
} (Region III, Fig. 1)

4.
Determine V
_{
ed
} using Eq. 11, V
_{
es
} and SV (Region III, Fig. 1)

5.
Simulate V
_{
lv
} using Eq. 5, Eq. 9, P
_{
ao
}, V
_{
es
} and V
_{
ed
} (Region III, Fig. 1)

6.
Calculate and normalise the TVE curve e(t) using Eq. 1
Analysis and validation
The proposed method was validated on experimentally gathered data. A range of input (P
_{
ao
}) and output (V
_{
lv
}, P
_{
lv
}) waveforms were continuously measured via catheter. This data allowed validation of individual model assumptions through comparison with directly measured output waveforms, as well as validation of the overall method through comparisons between the TVE curve as calculated using simulated and directly measured V
_{
lv
} and P
_{
lv
} wave forms.
The data set encompasses 46,318 heartbeats across 5 Piétrain pigs. A diverse clinical protocol provides the ability to assess intra and intersubject variability across a large amount of invasive and noninvasive measurements. Together they enable rigorous assessment and validation of the method.
Experimental procedure
The experimental protocol was approved by the Ethics Commission for the Use of Animals at the University of Liège, Belgium. Five male, pure Piétrain pigs weighing between 18.5 and 29 kg were sedated, anaesthetised and mechanically ventilated (GE Engstrom CareStation) with a baseline positive endexpiratory pressure (PEEP) of 5 cmH_{2}O (Fig. 4, Additional file 1: Table S1). Proximal aortic pressure was continually sampled using a pressure catheter (Transonic, NY, USA) with a sampling rate of 250 Hz. To provide direct measurements of P
_{
lv
} and V
_{
lv
} for validation, the heart was accessed via a median sternotomy, and an admittance pressure–volume catheter (Transonic, NY, USA) with a sampling rate of 250 Hz inserted into the left ventricle via an apical stab [36, 37].
To demonstrate a diverse range of cardiac states, several procedures were performed:

A single infusion of endotoxin (lipopolysaccharide from E. Coli, 0.5 mg/kg injected over 30 min) to induce septic shock. Septic shock drives a change in afterload conditions and is associated with a large variety of effects including an inflammatory response and capillary leakage that may lead to hypovolemia, decreased cardiac output, decreased ejection fraction and cardiac failure [38].

Several PEEP driven recruitment manoeuvres (RMs), both pre and postendotoxin infusion. RMs drive a change in preload conditions and are typically associated with a decrease in mean blood pressure and cardiac output [39].

One to four infusions of 500 mL saline solution over 30 min, pre and postendotoxin infusion, simulating fluid resuscitation therapy, a key component of hemodynamic resuscitation in patients with severe sepsis, which itself results in a change in circulatory volume [40].
Validation of significant model assumptions
Two major assumptions made in deriving this model are:

1.
That V
_{
d
} can be expressed as a function of baseline V
_{
es
}, as in Eq. 4

2.
That E
_{
es
} can be expressed as a function of HR, as in Eq. 8
Direct evaluation of the tracking of V
_{
es
} using different forms of ESPVR allows validation of both of these assumptions. In particular, 3 different methods of tracking V
_{
es
} were compared:

1.
Fixed
E
_{
es
}
and neglected
V
_{
0
}: The standard ESPVR (Eq. 6) with V
_{0} = 0 (a commonly used assumption [17, 18, 23])

2.
Fixed
E
_{
es
}
and fixed
V
_{
0
}: The standard ESPVR (Eq. 7) with V
_{0} = V
_{
d
} (allowing assessment of the validity of Eq. 4)

3.
Dynamic
E
_{
es
}
and fixed
V
_{
0
}: The ESPVR as used in the proposed method (Eq. 8) with V
_{0} = V
_{
d
}, and E
_{
es
} as a function of HR (allowing assessment of the validity of Eq. 8)
Validation of overall model
The overall method presented here is designed to simulate the TVE curve beatbybeat, without requiring invasive instrumentation of the heart or realtime imagebased monitoring, neither of which is clinically or ethically feasible in care. As such, validation of the method relies on comparison of the simulated TVE curve to the invasively measured, ‘true’ TVE curve, which is calculated using the catheter measured V
_{
lv
} and P
_{
lv
} waveforms for a single beat. This comparison is achieved by calculating the absolute and signed ‘error area’ between the measured and simulated TVE curve, according to Eqs. 12 and 13:
$$\varepsilon_{abs} = \frac{{\mathop \smallint \nolimits_{t = 0}^{1} \left {e_{sim} (t)  e_{meas} (t)} \right}}{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{meas} (t)} \right)}}$$
(12)
$$\varepsilon_{\text{sgn}} = \frac{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{sim} (t)  e_{meas} (t)} \right)}}{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{meas} (t)} \right)}}$$
(13)
where ε
_{
sim
} and ε
_{
meas
} are the simulated and measured TVE curves respectively, t is normalised time set to 1 for every heart beat to enable comparison over different beats, and ε
_{
abs
} and ε
_{
sgn
} and denote the absolute and signed errors respectively. An example TVE curve with an absolute error of 7.8% and bias of −3.4% is shown in Fig. 5, where shading denotes the error area.