A new electric method for non-invasive continuous monitoring of stroke volume and ventricular volume-time curves

Relative differential measurement

For every point in time during each cardiac cycle, it is to be established (by an algorithm) how much of the measured voltage differences is caused by volume changes in the atria, and how much by volume changes in the ventricles.

The design of the multi-electrode configuration needed to be optimized for this task. Therefore, we formulated the following two demands, which were to be met by the multi-electrode configuration:

(i) insensitivity to conductivity changes or diameter changes, during each single cardiac cycle, within vertical elongated structures like the aorta or other larger blood vessels

(ii) insensitivity to thorax dimensions and other factors that influence the average, global, current strength inside the thorax.

Demand (i) is met if the electrodes are placed in a vertical array over the central sternal line, and the voltages are measured between two adjacent electrodes with a relatively small inter-electrode distance. See the first five measuring electrodes (e0 to e4) in Figure 1. (The other electrodes, e5 and e6, are placed to the left and the right of the vertical sternal line, and used to deal with inter-patient anatomical differences, as explained later on). The measuring electrodes are numbered e ₀ to e ₆, and, for each value of k (with k∈{1…6}), the voltage ${\Phi }_{\mathrm{meas}}^{\left(k\right)}$ is measured between the two electrodes e _k and e _k−1. This results in a measurement of an approximation of a vertical spatial “differential” of the voltage distribution on the thoracic skin along the vertical sternal line. This “differential” along the vertical axis causes the measured voltage to be relatively insensitive to conductivity and diameter changes that are spread out in the vertical direction, as is the case with large vertical blood vessels.

As has been indicated in the introduction, the basis of the input of our system is formed by the voltage changes during a specific cardiac cycle, as function of the time τ(in milliseconds), with respect to the beginning of that cardiac cycle at τ=0, in which the beginning of a cardiac cycle (τ= 0) is defined as the moment in time when the R-peak of the ECG has its maximum. The measured voltage change between electrode pair k, as function of τ, therefore reads: ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(\tau \right)-{\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(0\right)$.

As will be shown in the Appendix, the voltage change ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(\tau \right)-{\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(0\right)$ is proportional to the strength of the incident applied current field near the heart. Furthermore, the ${\Phi }_{\mathrm{meas}}^{\left(k\right)}$ itself is proportional to the incident applied current field (near electrodes e _k and e _k−1), for all values of τ(including τ=0).

Therefore, the strength of the applied current is factored out due to the quotient in eq.(2), and the dimensionless numbers ${\phi }_{\mathrm{meas}}^{\left(k\right)}\left(\tau \right)$ are insensitive to variations in applied global current density.

Algorithm for retrieving ventricular volume-time curves and stroke volume

In ex-vivo calibration experiments on human postmortem hearts (to be described in more detail in section Methods: ex-vivo post-mortem calibration), we measured ${\Phi }_{\mathrm{ventric},\mathrm{before}}^{\mathrm{postmortem}}$ and ${\Phi }_{\mathrm{ventric},\mathrm{after}}^{\mathrm{postmortem}}$ values on a post-mortem thorax, in which ${\Phi }_{\mathrm{ventric},\mathrm{after}}^{\mathrm{postmortem}}$ is measured after an artificial increase of the volume (with a known fixed standardized volume ${\tilde{V}}_{\mathrm{calib}}$) in exclusively the ventricles of the post-mortem heart, and ${\Phi }_{\mathrm{ventric},\mathrm{before}}^{\mathrm{postmortem}}$ is the initial value before the artificial increase.

in which the index (k) refers to exactly the same locations on the thorax as the positions of the electrodes e _k in the in-vivo situation in eq.(2).

In order to retrieve the stroke volume and volume-time curves from the measured v _meas (τ), the HCP has to solve the value of ψ(τ)from eq.(7) for each point in time τ at which a measurement of v _meas has taken place (every 5 milliseconds in our HCP prototype).

The last equation (eq. (7)), together with the method to solve the volume-time curves (ψ(τ)) from eq. (7) using the measured v _meas (τ), constitute the core of the new VFR method presented in this paper.

Therefore, equation (7) has to be accounted for in the form of a fundamental physical reasoning that justifies this equation. This reasoning is provided in the Appendix (A1).

Furthermore, fundamentally speaking, any calculation of voltage changes on basis of volume changes of blood-filled lumina such as the ventricles, is a non-linear problem. Therefore, the limits of the accuracy of this linear equation will be examined in the Appendix as well (in A2).

A method for retrieving ψ(τ)from equation 7 is described directly below.

The “weight vectors” W _A and W _V are time-independent and completely based on only the ex-vivo “fingerprint” vectors f _A and f _V . As a result, the ψ(τ)is a linear function of the six measured voltages in the v _meas (τ)vector in eq.(10).

On basis of the results of the ex-vivo post-mortem calibration experiments (that will be described in section Methods: ex-vivo post-mortem calibration), a fixed known standard ventricular volume increase ${\tilde{V}}_{\mathrm{calib}}$ has been linked to the “fingerprint vector” f _V . Therefore, as an extreme and purely theoretical example, consider a case in which the volume of the atria would not change at all, and in which at some moment in time τ, the v _meas (τ) would be equal to f _V . Then the value of ψ(τ)would be equal to one (as can be seen in eq.(7)), and the ventricular volume change V(τ)−V(0) (in which V(τ)denotes the ventricular volume at time τ during a specific cardiac cycle, and V(0)denotes the ventricular volume at the beginning (τ=0) of that specific cardiac cycle) would be equal to ${\tilde{V}}_{\mathrm{calib}}$ at time τ.

This equation constitutes the final result that has been implemented in the algorithm of the experimental prototype of the HCP, in order to calculate the ventricular volume-time curve V(τ)−V(0)from the v _meas (τ)measured in-vivo on volunteers.

Experimental prototype of HCP

We developed a measuring device that consists of eight separate units, each containing a current source and an amplifier-demodulator [17, 18]. In this study, only six of these units were used. Each current source produces an alternating current of 1 mA _pp (milliAmperes peak-to-peak) at a distinct frequency in the range between 58 kHz and 90 kHz. The current sources are added so as to develop an electric field in the thorax containing components of different frequencies each to be measured by the according amplifier- demodulator. The current sources were located at approximately 2 meters from the volunteers in a patient-safe metal “front-end” rack system. The maximum applied current will not exceed 8 mA _pp which is well within the limits determined by the standard IEC-601-1, sub-clause 8.7.3. Each measuring channel contains an ECG amplifier that simultaneously detects the ECG as presented at the electrodes for measurement of the electric field that is developed by the injected current. In this manner 12 outputs are available: 6 outputs related to the voltages at a certain site on the thorax as a result of the injected currents and another 6 outputs presenting the ECG at this site. Moreover, each output containing data from the field developed by the injected current is split into two frequency ranges: a low-pass section and a band-pass section. The low-pass section with a cut-off frequency of 0.072 Hz produces an average voltage, V _avg , whereas the band-pass section produces an output in a bandwidth from 0.072 Hz up to 34 Hz. The latter output is amplified giving a voltage, V _fluct , rendering the instantaneous fluctuations in V with respect to the basic level V _avg . The current source and input amplifier of each unit are properly isolated according to the standards for body floating application to ensure safe operation when applied on patients. All data is digitized at a sampling rate of 200 Hz and made available through a serial communication interface that leads to a personal computer (PC). The system is equipped with two PC’s. During a time interval of exactly 20 seconds (a 20 seconds “measurement period”), PC #1 collected real time voltage data from the filter/amplifier unit. Directly after this interval of 20 seconds, there is a short interval of 4.3 seconds during which PC #1 collects no input data, but sends the compressed data buffered from the last “measurement period” on to PC #2. Immediately after that, PC #1 starts with the next “measurement period” of 20 seconds, during which PC #2 processes the data it just received from PC #1, i.e. PC #2 applies the VFR algorithm and displays the resulting ventricular volume-time curves, as well as the average Stroke Volume associated with the last measurement period of 20 seconds. The time that PC #2 needed for applying the algorithm and displaying the result was 14.8 seconds. Therefore, the “repetition time” of the total system (i.e., the time that lapsed between two subsequent updates of the displayed Stroke Volume) was 24.3 (= 20 + 4.3) seconds. Furthermore, the maximum delay between actual measurement of data (at the beginning of a measurement period of 20 seconds) to displaying the results is 39.1 (= 20 + 4.3+ 14.8) seconds.

Respiration-gated algorithm

In order to exclude artefacts in the measured voltage change pattern (${\phi }_{\mathrm{meas}}^{\left(k\right)}$ with k∈{1,2,…,6}) on the thoracic skin due to distortions caused by changing air volumes inside the lungs, we adopted a respiration-gated approach. Preliminary measurements have shown that the absolute value of ${\Phi }_{\mathrm{meas}}^{\left(k\right)}$ (as opposed to the dimensionless relative difference ${\phi }_{\mathrm{meas}}^{\left(k\right)}$ ) is a reliable indicator of the volume of air inside the lungs. Therefore, by monitoring the respiratory fluctuation in the series of absolute values of Φ _meas (0)from subsequent heart cycles, the HCP automatically selects a subset of cardiac cycles, out of the total set of cardiac cycles within a period of 20 seconds, that corresponds to a specific phase in the respiratory cycle (on basis of the trend in the series of ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(0\right)$ values), as well as to the same volume of air inside the lungs (on basis of the absolute value of ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(0\right)$ values). In this way, this “Φ-gated” algorithm serves as a respiration-gated algorithm. This selection procedure on basis of the ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(0\right)$ values takes place before the algorithm (as specified in eq.(12)) is applied.

G-suit deflation, and LVot Doppler as a reference technique

In order to cause changes in stroke volume in volunteers, and at the same time let other (environmental) parameters remain the same as much as possible, we used inflatable antishock trousers (known as “G-suits” to air force pilots). Such antishock trousers contain inflatable bladders inside the trousers. When inflated, these bladders produce pressure forces onto the legs, which prevent the blood to accumulate inside the legs when the airplane makes a sharp turn; hence these trousers prevent the pilot from loosing consciousness. It has been shown [19] that, in a standing position, sudden deflation of an inflated G-suit can cause the stroke volume to drop suddenly. We performed a procedure on 7 volunteers, in which a G-suit (Anti-G-Garment, CUTAWAY, CSU-13B/P, USAF, USA) was inflated gradually from 0 mmHg to 70 mmHg during 2 minutes and 30 seconds. Subsequently, a constant pressure of 70 mmHg pressure was maintained during 2 minutes, after which, finally, a sudden deflation back to 0 mmHg took place within 5 seconds.

The 7 volunteers were all healthy males, with a Body Mass Index ranging from normal to slightly overweight. The volunteers had no visible abnormalities in their thoracic anatomy, and were randomly selected from the male population on the Utrecht University Campus. All volunteers have read and signed an Informed Consent form, prior to the experiments. All procedures concerning the volunteers in this study have been executed in compliance with the guidelines of the University Medical Center Utrecht (and in compliance with the Helsinki Declaration), and have been approved by the Authority for Safety of the University Medical Center Utrecht.

Simultaneously, during the entire procedure, continuous HCP recordings, as well as LVot (Left Ventricular outflow tract) Doppler recordings were performed, using a Philips CX50 echocardiograph (version 2.0, Philips Medical Systems, Eindhoven, The Netherlands) with a S5-1 ultrasound transducer. Stroke volume was calculated as the Velocity Time Integral (VTI) multiplied by the LVot area. Pulse contour analysis was done offline for Velocity Time Integral (VTI). VTI was traced by hand and calculated by the echocardiograph on 1-3 curves in the each LVot registration. LVot registrations where made sequential with intervals of 30 sec. at rest, 20 sec during inflation-inflated and 3-4 sec. during deflation. Each registration holds 3 to 5 flow curves depending on the volunteer’s heart rate. In the apical 5-chamber view the Pulsed Wave Doppler sample volume was positioned about 0.5 cm proximal of the Aorta-valve in line with the blood flow direction as indicated by the color Doppler. This position was regularly checked, depending on the movements of the volunteer. The LVot area was calculated using the measured LVot diameter, average of 2 measurements, in a enlarged view of the Aorta-valve in the para-sternal long axes view.

This validation focussed on investigating the capability of monitoring changes in SV during de G-suit manipulations correctly. For each individual volunteer, a single and constant scaling factor K _volunteer was applied to the SV readings to let the average initial SV from the HCP coincide with the SV from the LVot Doppler.

results of the ex-vivo post-mortem calibration

For each of the electrodes in the electrode matrix shown in Figure 2, the voltages ${\Phi }_{\mathrm{ventric},\mathrm{before}}^{\mathrm{postmortem}}$ and ${\Phi }_{\mathrm{ventric},\mathrm{after}}^{\mathrm{postmortem}}$ have been measured, as well as the voltages ${\Phi }_{\mathrm{atrial},\mathrm{before}}^{\mathrm{postmortem}}$ and ${\Phi }_{\mathrm{atrial},\mathrm{after}}^{\mathrm{postmortem}}$, in which “before” and “after” refer to the moment in time with respect to the artificial filling procedure. All voltages were measured with respect to a reference electrode at the incisura jugularis (not shown in the figures). Two-dimensional linear interpolation has been used to create the continuous two-dimensional interpolations ${\Phi }_{\mathrm{ventric},\mathrm{before}}^{\mathrm{postmortem}}(x,z)$ and ${\Phi }_{\mathrm{ventric},\mathrm{after}}^{\mathrm{postmortem}}(x,z)$, as well as ${\Phi }_{\mathrm{atrial},\mathrm{before}}^{\mathrm{postmortem}}(x,z)$ and ${\Phi }_{\mathrm{atrial},\mathrm{after}}^{\mathrm{postmortem}}(x,z)$ from the discrete sets of measurements from the electrode matrix, in which the origin (x=0 and z=0) coincides with the position of the top end of the sternal bone, near the incisura jugularis, and x and z denote the horizontal and vertical position with respect to the origin, respectively.

The 2D color contour plots of $m_{\mathrm{ventric}}^{\mathrm{postmortem}}(x,z)$ and $m_{\mathrm{atrial}}^{\mathrm{postmortem}}(x,z)$ are rendered at the left side and the right side of Figure 3, respectively.

The positions of five electrodes (e ₁to e ₅) are indicated in Figure 3 by small O-shaped symbols.

The “+15” and “-15” in the denominator of eqs. (13) and (14) indicate that the normalization of the m-values in these plots are based on the inter-electrode distances (in the vertical direction) between e ₁, e ₂, and e ₃ (viz: 30 mm).

for (k) ∈{2,3}.

The ${\mu }_{\mathrm{ventric}}^{\left(k\right),\mathrm{postmortem}}$ of the other electrode pairs are calculated analogously, taking into account other inter-electrode distances. The ${\mu }_{\mathrm{atrial}}^{\left(k\right),\mathrm{postmortem}}$ are calculated in the same way, on basis of the $m_{\mathrm{atrial}}^{\mathrm{postmortem}}(x,z)$ in eq.(14).

Results of the in-vivo tests of the HCP on human volunteers

In Figure 4, typical examples are rendered of ventricular volume-time curves, produced by the HCP, of a volunteer in rest, in a standing position, before inflation of the G-suit, in which the “ventricular volume change” (i.e., V(τ)−V(0)) was calculated using eq. 12.

An example of the simultaneous SV readings from HCP and LVot Doppler, as function of time, during inflation and deflation of the G-suit is rendered in Figure 5.

The combined results of all 7 volunteers are rendered in Figure 6 (Linear Regression) and Figure 7 (Bland Altman plot).

The Pearson correlation coefficient in Figure 6 was R=0.892.

In the Bland-Altman plot, the mean bias was -1.6 ml, with 2SD = 14.8 ml. The upper limit of agreement was 13.2 ml, and the lower limit of agreement was -16.4 ml.

Ex-vivo post-mortem measurements

The ex-vivo post-mortem measurements show that the spatial 2D distribution of voltage changes over the frontal thoracic skin due to ventricular emptying (the ventricular “fingerprint”) is clearly different from the atrial “fingerprint”.

A number of features can be discerned in the experimental results in Figure 3. In the atrial fingerprint at the right side of Figure 3, a yellow (negative) blob (minimum value of about -0.03) centered around (-30,-70) is visible, as well as a positive counterpart (purple, positive blob, maximum value of about 0.03) near (30, -180). In the Appendix, and more particularly in Figure 8, it was predicted that the normalized voltage change distribution on the thoracic skin (fingerprint) in the post-mortem ex-vivo experiments should have approximately the same appearance as if the heart were replaced by a set of electric dipoles. The presence of the yellow (negative) blob and the purple, positive blob in the atrial plot in Figure 3 are in accordance with a “dipole-like field” from a set of dipoles located inside the heart.

In the ventricular fingerprint (at the left side of Figure 3), it can be seen that the entire “dipole-like field” is positioned much lower (i.e., displaced in the caudal direction with respect to the atrial “dipole” with a distance of about 120 mm). For reasons explained in the Methods section, the post-mortem ex-vivo measurements were confined to the thoracic skin. Therefore, only the upper part of the top blob (orange, negative) was located on the post-mortem thorax in the ex-vivo ventricular fingerprint measurements, at about (30,-180) in the left panel in Figure 3.

In-vivo tests on human volunteers

The shapes of the ventricular volume-time curves (see Figure 4) were consistent with the volume-time curves from the literature. The realistic shapes of the ventricular volume-time curves produced by the HCP during the tests on human volunteers indicate that the atrial and ventricular fingerprints are indeed different enough to enable the HCP algorithm.

Furthermore, the “relative differential measurement” method, in which electrodes are placed relatively close to each other along the vertical central sternal line (as explained in subsection relative differential measurement), seems to be effective in suppressing the influence of conductivity changes in larger vertical blood vessels inside the thorax. (Or, more precisely: the relative differential measurement method is relatively insensitive to changes in diameter and blood conductivity (which varies with blood velocity) that occur within vertical elongated conductors like the aorta, vena cava, and pulmonary arteries during the cardiac cycle).

One volunteer (volunteer ♯ 8) was excluded from the validation section because LVot Doppler incidently detected a leakage of the mitral valve. Interestingly, the HCP has produced a volume-time curve for this volunteer that is distinctly different from the volume-time curves of the other volunteers, and, moreover, this particular volume-time curve was assessed by cardiologists to be typical for mitral valve leakage. Further research is warranted to investigate if the HCP not only can produce reliable stroke volume values, but also may be used to determine particular patterns in the volume-time curves to diagnose e.g. diastolic dysfunction, cardiomyopathy or valvular disease.

Therefore, the ventricular volume-time curves have the potential of being a useful diagnostic monitoring tool as well.

Comparison of the VFR with other techniques

In contrast to Impedance Cardiography (ICG), the VFR method uses multiple electrodes to create input for 6 independent and simultaneous voltage input channels. Our method is not based on impedance cardiography (ICG) algorithms, since we do not calculate any impedance measure, but use spatial patterns to retrieve volume changes of specifically the ventricles.

Some ICG-like techniques (such as e.g. Woltjer et al [20]) use multiple electrodes, but - in contrast to our method - still measure only one single voltage, because the leads of many of these electrodes are interconnected to form a single input channel.

Comparison of the bias and precision (2SD) of the prototype of the HCP in this study with that of other techniques as listed in e.g. [3], shows that the HCP yielded satisfactory results as a completely non-invasive technique.

Sources of inaccuracy

In the appendix, in Figure 9, three major sources of non-linearity are depicted schematically. Using the formula (eq.(34)) derived in the appendix, confines to the maximal deviations from linearity have been calculated, yielding the graphs rendered in Figure 10.

For a volume change that equals two times the calibrated initial volume change, the non-linear effects may cause a voltage change that is maximally 14% higher than the linear approximation. Using the linear approximation therefore leads to an overestimation of the stroke volume of maximally 14% in this case.

For a volume change that equals half of the calibrated initial volume change, the non-linear effects may cause a voltage change that is maximally 6% lower than the linear approximation. In this case, the linear approximation therefore leads to an underestimation of the stroke volume of maximally 6%.

A1: fundamental relation between measured voltages and volume changes

In this subsection, the validity of the assumed linearity in eq.(7) is examined. To this end, we first focuson only the ventricles and assume, for the moment, that the atria do not exist, and examine to what extent the measured changes in voltages, as combined into the vector v _meas , may be considered to be proportional to the volume change V(τ)−V(0). Furthermore, we concentrate on the stroke volume of each heart beat, and enumerate the heart beats using an index (n) (n∈{1,2,3,…}), and define ${\tilde{V}}_{\left(n\right)}$ as the stroke volume of beat number n. Let v _meas,(n)be defined as the v _meas vector at the moment in time of maximum volume change during cardiac cycle number n (i.e., the end-systolic moment in cardiac cycle number n), and let the index (0) refer to the first heartbeat after the start of the measurements (baseline condition, n=0). Evidently, in the course of time after n=0, during later heartbeats, the stroke value may differ from the initial stroke value ${\tilde{V}}_{\left(0\right)}$

in which λ _(n)is a scalar (representing the proportionality factor for cardiac cycle number n), and in which the 6 values ${\phi }_{\mathrm{meas}}^{\left(k\right)}$ (k∈{1,2,…,6}) in v _meas are caused by volume changes in the only volumes present, i.e. the ventricles. The assumption $\mathcal{L}$thus reads that if during cardiac cycle # (n) the stroke volume ${\tilde{V}}_{\left(n\right)}$ equals λ _(n) times the initial stroke volume ${\tilde{V}}_{\left(0\right)}$, then all 6 voltages in v _meas,(n)would equal λ _(n)times the initial voltages during cardiac cycle # (0) as well.

Evidently, based on the general fact that calculation of voltages from conductivity distributions is a non-linear problem, it is clear on beforehand that the assumption $\mathcal{L}$will not be true with exact precision; hence the ≈-sign in eq.(16)

In the experiments on human volunteers, an initial calibration procedure using Doppler Ultrasound has been performed at the start of the experiment, and hence the value of the actual initial ${\tilde{V}}_{\left(0\right)}$ has been provided by Doppler Ultrasound for each volunteer.

The assumption $\mathcal{L}$can only be valid (approximately) if, for any value of n>0, the shape change of the volume, during the volume change during cardiac cycle number n, is not completely deviant from the shape change of the volume during n=0. In other words, we make the assumption that sudden ruptures or other sudden abnormal patterns of volume changes are excluded. We will refer to this assumption as the “assumption of regularity $\mathcal{R}$”, and will specify this assumption $\mathcal{R}$ more precisely later on in this subsection.

In order to assess the limits of accuracy of the assumption $\mathcal{L}$, a fundamental relation between the measured relative voltage changes φ and volume changes of blood-filled lumina is derived below, starting with explaining the nature of the applied current field.

The oscillation frequency f of the measured AC voltages, which equals the oscillation frequency of the current injected into the patient via the current feeding electrodes on the legs and arms or head, is high in comparison to the physiological processes like respiration or the motion of the heart during the cardiac cycle. Furthermore, the frequency f is low enough to let the values of 2Πfε (in which ε is the dielectric permittivity) be negligible with respect to the differences in specific conductivity σ for the relevant tissues [22]. Therefore only the amplitude of the measured AC voltages needs to be considered; in our set-up each ${\Phi }_{\mathrm{meas}}^{\left(k\right)}\left(\tau \right)$ value was defined as the amplitude of the oscillating voltage measured between the electrodes e _k and e _k−1 at time τ. This amplitude of the voltage was established every 5 milliseconds.

Directly below, a relation between the measured voltage changes φ and volume changes of blood-filled lumina is derived, on basis of the law of conservation of charge (the continuity equation) and Ohm’s law.

in which x is a three-dimensional spatial coordinate ( i.e. x denotes a point within 3D space), and J(x) denotes the current density at point x, and $\mathcal{T}$ denotes the complete thorax, including the thoracic skin, and everything inside the thorax.

Let the 3D vector field −∇log(σ(x))be denoted as s(x).

This is an implicit equation for E(x) for a given s(x) distribution. For a model of the human anatomy in which a limited number of tissue types are defined (such as “blood”, “muscle tissue”, “bone”, etc.), the spatial conductivity distribution σ(x) is piecewise constant; i.e. the value of s(x)is zero everywhere, except at the boundary between two different tissue types. For a sharp boundary, we have [21]: $\mathbf{s}\left(\mathbf{x}\right)=2\widehat{\mathbf{n}}\frac{{\sigma }_1-{\sigma }_2}{{\sigma }_1+{\sigma }_2}$, in which $\widehat{\mathbf{n}}$ is the normal unit vector perpendicular to the boundary surface at point x, and σ ₁and σ ₂ are the specific conductivities of the two tissue types that meet at the boundary surface.

Since $\nabla ·\mathbf{E}=\frac{\rho }{\epsilon }$ (according to the Maxwell equations, in which ρ is the charge density and ε is the dielectric constant), the right-hand side of eq.(20), i.e. E(x) · s(x), may be regarded as a charge density divided by the dielectric permittivity ε. Therefore, points within the thorax having non-zero values of s(x)may be considered as equivalent “secondary” sources, responsible for non-zero divergence of E within the thorax at points on boundary surfaces. The effect of these secondary sources on the rest of the thorax (including the measuring electrodes on the thoracic skin) may be calculated using the Green function G of the Poisson equation.

This is still an implicit equation from which $\stackrel{\mathbf{~}}{\mathbf{E}}$ can not be calculated directly.

In eq.(21), the first term (${\mathbf{E}}_0\left(\mathbf{x}\right)·\tilde{\mathbf{s}}(\mathbf{x},\tau )$), indicated by (⋆), represents the first order approximation, whereas $\tilde{\mathbf{E}}(\mathbf{x},\tau )·\mathbf{s}(\mathbf{x},\tau )$, indicated by (⋆⋆), represents the self-reference in this equation, which leads to higher-order corrections in the form of a perturbation series if an exact expression for $\stackrel{\mathbf{~}}{\mathbf{E}}$ is derived. In the appendix, the exact solution to eq.(21) is presented in the form of such a perturbation series.

Since the purpose of this subsection is to determine to what degree of accuracy the actual volume change V(τ)−V(0) may be considered to be proportional to the ψ(τ)(as calculated from the measurements using eq. (11)), we first examine this proportionality on basis of the first term (${\mathbf{E}}_0·\tilde{\mathbf{s}}\left(\tau \right)$) alone, and, subsequently, calculate the higher-order deviations from this proportionality on basis of the perturbation series that constitutes the solution to eq.(21) as a whole.

in which ∫ _thorax d³ ξ denotes integration over 3D space. with ξ denoting points in 3D space. From this equation, a basic relation between voltage changes and volume changes can be derived, as is indicated in Figure 8.

Consider, as a simplified example, the boundary of a single volume of blood within an environment of tissue. This boundary is rendered in Figure 8a for the initial situation at τ=0. The value of s(x,0)is zero everywhere, except at the boundary. At a later instant τ _ES during the same cardiac cycle (in which the subscript “ES” refers to “end-systole”), the volume of blood has reached its maximum volume change.

The model depicted in Figure 8 can incorporate volume changes in a direction perpendicular to ${\hat{\mathbf{e}}}_0$ as well: If the volume is becoming smaller in a direction perpendicular to ${\hat{\mathbf{e}}}_0$ , then some rods near the boundary become gradually shorter until they vanish completely (when the length of the rod becomes zero). If the volume is becoming larger in a direction perpendicular to ${\hat{\mathbf{e}}}_0$ , some vanished rods with length zero obtain non-zero length again (effectively adding new rods at the boundary of the volume).

Similarly, the charges s·E ₀(x)may be considered to be a set of dipoles ${\mathbf{p}}_i=q_i{h}_i{\hat{\mathbf{e}}}_0$ (as explained in Figure 8), enumerated by the index i, with dipole length h _i and charge q _i .

We will now examine the proportionality between the volume change ${\tilde{V}}_{\left(n\right)}=V\left({\tau }_{\mathrm{ES}}\right)-V\left(0\right)$ and the voltage differences measured on the thorax resulting from the set of dipoles ${\mathbf{p}}_i=q_i{h}_i{\hat{\mathbf{e}}}_0$.

in which ${\mathbf{G}}_{\mathrm{dipole}}\left(\mathbf{x}\right)=\widehat{\mathbf{x}}/\left(4\mathrm{\Pi \epsilon }\right(\mathbf{x}·\mathbf{x}\left)\right)$.

from which it becomes apparent that, having used the assumption of regularity $\mathcal{R}$(eq. (25)) and the first order approximation in eq.(22), ${\Phi }_{\left(n\right)}^k\left({\tau }_{\mathrm{ES}}\right)-{\Phi }_{\left(n\right)}^k\left(0\right)$is indeed proportional to ${\lambda }_{\left(n\right)}\sum _i{h}_{i,\left(0\right)}{A}_i$, and hence assumption $\mathcal{L}$holds in this case. Furthermore, eq.(27) shows that ${\Phi }_{\left(n\right)}^k\left({\tau }_{\mathrm{ES}}\right)-{\Phi }_{\left(n\right)}^k\left(0\right)$is proportional to the incident field strength E ₀as well.

As has been indicated before however, it is clear on beforehand that the assumption $\mathcal{L}$ will not be true with exact precision, due to the fact that the relation between volumes and voltages in fundamentally non-linear. The strength of the non-linear effects will be examined in the next subsection (A2).

A2: non-linear effects

In order to transform equation (28) into an explicit equation for $\stackrel{\mathbf{~}}{\mathbf{E}}$ (and hence solving eq.(29) as well), an iterative substitution method is used that is comparable to the iterative substitution that produces the Born-Neumann series from the Lippmann-Schwinger equation, i.e.: the symbol $\stackrel{\mathbf{~}}{\mathbf{E}}$ appearing in the right-hand side of eq.(28) is replaced by the expression that is formed by the entire right-hand side of eq.(28). This results in a new equation in which $\stackrel{\mathbf{~}}{\mathbf{E}}$ appears again near the end of the right side of the resulting equation, and hence the substitution procedure can be repeated, which yields an iterative substitution procedure.

in which, again, the symbol ∗denotes the convolution operator in three dimensions.

An important parameter for the assessment of convergence of the series $\mathcal{A}$in eq. (30) and eq.(31)) is the fraction $\frac{\mid \stackrel{\mathbf{~}}{\mathbf{E}}\left(\mathbf{x}\right)\mid }{\mid {\mathbf{E}}_0\left(\mathbf{x}\right)\mid }$.

If the fraction $\frac{\mid \stackrel{\mathbf{~}}{\mathbf{E}}\left(\mathbf{x}\right)\mid }{\mid {\mathbf{E}}_0\left(\mathbf{x}\right)\mid }$ is small compared to one, then the higher order (large ϑ) terms in (31) become rapidly irrelevant with increasing ϑ.

Since the cardiac motion causes only a small distortion of the applied current field, we have that $\frac{\mid \stackrel{\mathbf{~}}{\mathbf{E}}\left(\mathbf{x}\right)\mid }{\mid {\mathbf{E}}_0\left(\mathbf{x}\right)\mid }$ is indeed small compared to one, as can be shown for the theoretical case of a single spherical volume representing a ventricle in a homogeneous surroundings. For this theoretical case, the maximum value of $\frac{\mid \stackrel{\mathbf{~}}{\mathbf{E}}\left(\mathbf{x}\right)\mid }{\mid {\mathbf{E}}_0\left(\mathbf{x}\right)\mid }$ outside the sphere can be calculated exactly, and equals 0.14 for the conductivity values of blood and muscle tissue, and is reached near the surface of the sphere on the central axis of the sphere parallel to the incident field E ₀. For most other positions x, the value of $\frac{\mid \stackrel{\mathbf{~}}{\mathbf{E}}\left(\mathbf{x}\right)\mid }{\mid {\mathbf{E}}_0\left(\mathbf{x}\right)\mid }$ is much lower.

We now use eq.(31) to examine the non-linear character of the relation between volume changes and the measured Φ^(k)(τ) − Φ^(k)(0) by adding various anatomical and cardiological features, starting with the above mentioned simple theoretical case of a single spherical volume of blood (with σ _blood ) inside an infinite homogenous surroundings (with specific conductivity σ _tissue ). From the literature [21] it is known that, in this specific case, the electric field in the surroundings of the sphere is exactly equal to the sum of the incident field plus the field from a single dipole located at the center of the sphere.

As a result, the Φ^(k)(τ) − Φ^(k)(0)will be exactly proportional to the volume of the sphere.

Non-linear effects however do occur in $\mathcal{A}$ in eq.(31) if other objects are present besides the single sphere. Evidently, the true deviation from linearity in a complex realistic anatomical model can not be rendered in an explicit analytical expression. However, it is possible to derive analytically a set of boundaries that confine the possible deviations from linearity to a well-defined and narrow range for each specific anatomical and cardiological feature. In order to do so, these specific anatomical and cardiological features are translated into sets of spheres (as will be explained below). First, let the change in the incident field on a sphere S, caused by the presence of another sphere S’, be denoted by $\breve{\mathbf{E}}\left(\mathbf{x}\right)$. In order to calculate the confines of the range of the possible deviations from linearity, the following step is taken: for each point on the surface of S, the vector $\breve{\mathbf{E}}\left(\mathbf{x}\right)$ of the incident field on S, due to the presence of the other sphere S’, is replaced by the same vector ${\breve{\mathbf{E}}}_{max}$, in which ${\breve{\mathbf{E}}}_{max}$ is the maximum (“worst case”) value of all original $\breve{\mathbf{E}}\left(\mathbf{x}\right)$ vectors on the surface of sphere S. This produces a stronger non-linearity than in the case of an exact calculation, and hence provides an analytical expression for the upper limit to the strength of the non-linear effects.

The following three fundamental anatomical and cardiological features are considered that each produce a specific deviation from linearity (see Figure 9):

(a) The non-spherical shape of the ventricles (Figure 9a);

(b) The proximity of the heart to a boundary surface (skin-air), viz. the frontal surface of the thoracic skin (Figure 9b);

(c) The cardiological phenomenon that, e.g. during the rapid filling phase of the ventricles, the volume changes in the atria tend to be comparable to the volume changes in the ventricles, but have the opposite sign. In a completely linear model, neglecting all higher order terms (i.e., neglecting the term indicated by (⋆⋆) in eq.(21) ), the volume changes in the atria would contribute to only α(τ)in eq.(7), and the volume changes in the ventricles would contribute to only ψ(τ). However, indirectly, a volume change in the atria will change also the way that a volume change in the ventricles is perceived by the measuring electrodes, because a volume change in the atria will change the strength of the incident field on the ventricles, giving rise to a second order effect. (Figure 9c)

As an example, consider the following calculation of an upper limit to the second order deviation from linearity for feature (b). In Figure 9b, the feature of the proximity of the heart to a boundary surface (skin-air), i.e. feature (b), is rendered schematically. The method of image charges [21] is used to replace the skin-air boundary by a mirror image of the heart in a homogenous continuum.

in which the last term, indicated by the third horizontal brace at the far right, is proportional to the square of λ _(n), because λ _(n) and ${\breve{\mathbf{E}}}_{max}$ are both proportional to λ _(n). As a result, the $\eta {\lambda }_{\left(n\right)}^2{E}_0$ under the third horizontal brace at the far right represents the second-order effect, which is a deviation from the first order linear relation between ${\Phi }_{\left(n\right)}^{\left(k\right)}\left({\tau }_{\mathrm{ES}}\right)-{\Phi }_{\left(n\right)}^{\left(k\right)}\left(0\right)$ and λ _(n).

Using known specific conductivities of tissues from the literature, and known typical anatomical distances and sizes, the value of η for feature (b) has been calculated, resulting in η=0.035. The corresponding graph of formula (34) using η=0.035is the dotted green line in Figure 10. The slender area between this dotted green line and the identity line (black solid line) in Figure 10, represents the area containing all possible deviations from linearity for feature (b).

Similarly, a graph was calculated for the entire perturbation series in eq.(31), using a geometrical power series (i.e.: $\sum _{\vartheta =0}^{\infty }{p}^{\vartheta }=1/(1-p)$ for any scalar p) for feature (b). This resulted in the continuous green line in Figure 10.

For each of the three features (a), (b) and (c) mentioned above, the η values have been calculated.

The value of η for the feature (a) is −0.033, and the combination of features (b) and (c), excluding feature (a) to create a worst case, yields η=0.142. The corresponding graphs are rendered in Figure 10 as well.

For a volume change ${\tilde{V}}_{\left(n\right)}$ that equals two times the initial volume change ${\tilde{V}}_{\left(0\right)}$, the non-linear effects may cause a voltage change that is maximally 14% higher than the linear approximation. Using the linear approximation therefore leads to an overestimation of the stroke volume of maximally 14% in this case.

For a volume change ${\tilde{V}}_{\left(n\right)}$ that equals half of the initial volume change ${\tilde{V}}_{\left(0\right)}$, the non-linear effects may cause a voltage change that is maximally 6% lower than the linear approximation. In this case, the linear approximation therefore leads to an underestimation of the stroke volume of maximally 6%.