A2: non-linear effects

In the subsection above, the first order explicit equation (22) has been derived from the general implicit equation (21), by neglecting the self-referential term indicated by (⋆⋆) in eq.(21). The exact version of equation (22) however, i.e. without neglecting the self- referential term indicated by (⋆⋆) in eq.(21), reads:

E(x) = ∇

and hence the voltage difference between the two electrodes e k and e k− 1 equals:

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

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

E appears again near the end of the right side of the resulting

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equation, and hence the substitution procedure can be repeated, which yields an iterative substitution procedure.

In the top line of eq.(30) the result of the iterative substitution is rendered, and sub- sequently “expanded” by distribution through parentheses (in the second line of eq.(30)), yielding a perturbation series:

E = ∇G ∗ (E 0 · s + s · ∇G ∗ (E 0 · s + s · ∇G ∗ (. . . . . .)))

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

The voltage difference between the two electrodes e k and e k− 1 hence reads:

An important parameter for the assessment of convergence of the series A in eq. (30) and eq.(31)) is the fraction E(x)| |E 0 (x)| .

If the fraction E(x)| |E 0 (x)| 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 E(x)| |E 0 (x)| 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 E(x)| |E 0 (x)| 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

E(x)|

0 . For most other positions x, the value of |E 0 (x)| is much lower.

We now use eq.(31) to examine the non-linear character of the relation between vol-

(k)

(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.

(k) ( 0) will be exactly proportional to the volume of the sphere.

(k)

Non-linear effects however do occur in 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 car-

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diological 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 E(x) ˘ . 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 ˘E(x) of the incident field on S, due to the presence of the other sphere S ′ , is replaced by the same vector ˘E max , in which ˘E max is the maximum (“worst case”) value of all original ˘E(x) 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 the spherical modeling, increasing the stroke volume (as described by s ) with a factor λ is equivalent to replacing s with λ s in its effect on the measuring electrodes. Furthermore, the ˘E(x) near sphere S, caused by the presence of the mirror sphere S ′ , is proportional to the volume of the mirror sphere S ′ , and therefore also proportional to λ, and, as a result, the maximum ˘E max is proportional to λ as well. Since | ˘E(x) | is

also proportional to | E 0 |, the | ˘E max | may be written as ληE 0 , in which η depends

on the distance between the spheres. As a result, on basis of eq.(29), the expression

(n) ( max 0) that has a maximum deviation from linearity, yields:

× E 0 ( ξ ) · (λ (n) s( ξ )) + (s 0 ( ξ)+λ (n) s( ξ )) · ˘ E max

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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 ˘E max are both proportional to λ ( n) . As a result,

the ηλ 2 ( n) E 0 under the third horizontal brace at the far right represents the second-order ( k) (τ )

( n) ( 0) and λ ( n) . Since ϕ k

( k) ( n) ( 0) (by definition, see eq.(2)) and

( n) ( 0) ∝E 0 , the three proportionalities indicated under the braces in eq.(32), i.e. ∝λ ( n) E 0 , ∝ ηλ ( n) E 0 ,

and ∝ ηλ 2

( n) E 0 , respectively, entail the following equation:

( n) (τ ES ) max = aλ ( n) + aηζ λ ( n) + aηλ ( n) (33) in which a is an unknown scalar, and ζ

= 2 2. For the special case n = 0, i.e. the start of

the measurements and the very cardiac cycle that the initial calibration refers to, we have: λ V ( n)

( 0) = 1, because by definition λ ( n) = V ( (see eq.(16)). As a result, the unknown scalar a 0)

V ( cancels out if we consider the ratio ϕ n)

( n) (τ ES )/ϕ ( 0) (τ ES ) as function of V ( 0) :

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 = 0.035 is 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.: ∞ ϑ =0 p ϑ = 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.

V ( n)

V ( 0) , the non-

linear effects may cause a voltage change that is maximally 14% higher than the linear

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approximation. Using the linear approximation therefore leads to an overestimation of the stroke volume of maximally 14% in this case.

V (n)

V ( 0) , 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 underestima- tion of the stroke volume of maximally 6%.

Abbreviations

CO = Cardiac Output; G-suit = antishock trousers (containing inflatable bladders inside the trousers); HCP = Hemodynamic Cardiac Profiler (i.e., the prototype of the new system described in this paper); LVot = Left Ventricular outflow tract; SV = Stroke Volume; VFR method = Ventricular Field Recognition method.

Competing interests

The authors declare that they have no competing interests. HG has received some reimbursements for advice concerning the technical setup of the experiments. Furthermore, HG, AO and CH have some financial interest in an organization that potentially may profit from publication of this manuscript in the future. HG, AO, and CH have however not been involved in the interpretation of results and conclusions in this paper.

Authors’ contributions

MK invented the Ventricular Field Recognition method, developed the mathematical formulation and derivations in the paper, and drafted the manuscript except for parts of the ex-vivo method and electronic hardware sections. HG has developed the electronics of the HCP, wrote the electronic hardware section and reviewed the entire manuscript. FR has made essential contributions to elucidating the need for separating atrial filling effects from ventricular filling effects. MR and TH have co-planned and executed the ex-vivo measurements, and contributed to the ex-vivo methods section. AO has initialized and organized the use of G-suit stroke volume manipulation in the in-vivo experiments. RR, FK, RB, PD, AO, CH and WB have contributed to the methods, planning, and execution of the in-vivo experiments. All authors have read and approved the final manuscript.

Acknowledgements

The authors wish to thank Willem van Wolferen, Simon Plomp, and Fiona van Zoomeren from the dept. of Anatomy of the University Utrecht for their outstanding support and kind cooperation. Furthermore, René van de Vosse, Ernest Boskovic and other co-workers from the dept. of Medical Technology, as well as Dr. Maarten J. Cramer from the dept. of Cardiology, are gratefully acknowledged for their excellent assistance.

Author details

MK, MR, PD, and TH are with the dept. of Medical Technology of the University Medical Center Utrecht (UMCU), Utrecht, the Netherlands; RR, RB, and WB are with the depts. of Cardiology, Anatomy, and Anesthesiology, respectively, of the UMCU, Utrecht, the Netherlands; HG is with Goovaerts Instruments, Kortenhoef, NL ; AO and CH are with Hemologic BV, Amersfoort NL; FK is with ESC Medical Technology, Winschoten NL; FR is with the University Hospitals Leuven, Leuven, Belgium.

Received: 7 February 2012 Accepted: 25 July 2012 Published: 17 August 2012

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Cite this article as: Konings et al.: A new electric method for non-invasive continuous monitoring of stroke volume and ventricular volume-time curves. BioMedical Engineering OnLine 2012 11:51.

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A2: non-linear effects