Introduction

The impedance cardiography is a detective technique used to measure the cardiac functions and to monitor the hemodynamic changes, and it has many advantages such as noninvasiveness, good reproducibility and simple measurement.1 , 7 , 13 However, the scientific basis of the impedance cardiography is not understood fully, so it has not been widely accepted by clinicians.5 A great deal of research has proven that the impedance cardiography measured with Kubicek’s method (or Sramek’s method) is a kind of the mixed impedance signal, which results from the volume changes of the aorta, blood vessels in the lung and ventricles.2 , 4 , 5 , 11 , 12 , 14 , 16 For this reason, we try to separate their impedance change components from the mixed impedance signals on the chest surface by means of the waveform reconstruction.

Since it is necessary to establish the thoracic impedance equations for the waveform reconstruction, we have to elucidate the mechanism of the formation for the thoracic impedance change (i.e. the impedance change on the chest surface). Two problems are involved in it: (1) which factors relate to the thoracic impedance change resulting from single blood vessel? (2) How to gather the thoracic impedance changes resulting from multiple blood vessels as a sum? In order to solve these problems, the theoretical analysis and experimental study are made in this paper trying to consummate the theories raised by the pioneer authors in this area.

Method

Relevant Knowledge

Measuring Principle of Impedance Cardiography

As shown in Fig. 1, in the measurement of Impedance cardiography, four electrodes (i.e. the current electrodes I 1 and I 2, the voltage detective electrodes E 1 and E 2) are generally arranged on the chest surface.7 The alternating current I 0, with a constant current in its root-mean-square (rms), enters into the human body through the current electrodes I 1 and I 2.

Figure 1
figure 1

Measured principle

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If Z 0 and ΔZ are the basal impedance and impedance change between the electrodes E 1 and E 2, respectively, the total impedance Z should be equal to Z 0 + ΔZ. The voltage U between E 1 and E 2 can be expressed as8

$$ \begin{aligned} U & = I_{0} \left( {Z_{0} + \Updelta Z} \right) \\ & = I_{0} Z_{0} + I_{0} \Updelta Z = U_{0} + \Updelta U \\ \end{aligned} $$

where U 0 and ΔU in the above equation correspond to I 0 Z 0 and I 0ΔZ, respectively, i.e.,

$$ U_{0} = I_{0} Z_{0} \quad \Updelta U = I_{0} \Updelta Z $$
(1)

It is evident that the voltage change ΔU between E 1 and E 2 is directly proportional to the impedance change ΔZ. Thus the impedance change can be obtained by means of measuring the corresponding voltage change.

Electrical Field Distribution in the Cylindrical Volume Conductor

The pane ABCD in Fig. 2 shows a cylindrical volume conductor with uniform conductivity. When the voltage U i is applied to the cylindrical volume conductor through the input electrodes V 1 and V 2, the electrical field distribution shown as the dotted lines appears in it. The voltage detective electrodes E 1 and E 2 are placed on the lateral of the cylinder to measure the output voltage U o. It has been proved by the experiment that U o is directly proportional to the input voltage U i of the volume conductor and the distance L between the detective electrodes E 1 and E 2, while it is inversely proportional to the distance d between the lines V 1 V 2 and E 1 E 2 (see Fig. 2),6 i.e.

Figure 2
figure 2

Cylindrical volume conductor

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$$ U_{\text{o}} = k{\frac{{U_{\text{i}} L}}{d}} $$
(2)

Here k is the proportional coefficient.

Addition of the Voltages

It is assumed that there are multiple point charges (q 1, q 2,…,q N ) in a conductor, and the A and B are two points in the electrical field caused by these charges together. The potentials caused by the q 1, q 2,…,q N at point A are V 1A, V 2A,…,V NA, respectively, while that at point B are V 1B, V 2B,…,V NB, respectively. According to the principle of adding the potentials,9 , 15 the potentials at A and B (V A and V B) can be expressed as, respectively

$$ \begin{aligned} V_{\text{A}} & = V_{{1{\text{A}}}} + V_{{2{\text{A}}}} + \cdots + V_{{N{\text{A}}}} \\ V_{\text{B}} & = V_{{1{\text{B}}}} + V_{{2{\text{B}}}} + \cdots + V_{{N{\text{B}}}} \\ \end{aligned} $$

From the above equations, the potential difference (V A − V B) between A and B can be obtained,

$$ \begin{aligned} V_{\text{A}} - V_{\text{B}} & = \left( {V_{{1{\text{A}}}} + V_{{2{\text{A}}}} + \cdots + V_{{N{\text{A}}}} } \right) - \left( {V_{{1{\text{B}}}} + V_{{2{\text{B}}}} + \cdots + V_{{N{\text{B}}}} } \right) \\ & = \left( {V_{{1{\text{A}}}} - V_{{1{\text{B}}}} } \right) + \left( {V_{{2{\text{A}}}} - V_{{2{\text{B}}}} } \right) + \cdots + \left( {V_{{N{\text{A}}}} - V_{{N{\text{B}}}} } \right) \\ \end{aligned} $$
(3a)

where (V 1A − V 1B), (V 2A − V 2B),…,(V NA − V NB) are potential differences between A and B produced by the q 1, q 2,…,q N , respectively. The potential difference called also voltage, and it is denoted by U in the present text. It is assumed that the U 1, U 2,…,U N are the voltages between A and B produced by the q 1, q 2,…,q N , respectively, and U is the total voltage between A and B. Thus

$$ \begin{aligned} U & = \left( {V_{\text{A}} - V_{\text{B}} } \right),\;U_{1} = \left( {V_{{1{\text{A}}}} - V_{{1{\text{B}}}} } \right), \\ U_{2} & = \left( {V_{{2{\text{A}}}} - V_{{2{\text{B}}}} } \right), \ldots ,U_{N} = \left( {V_{{N{\text{A}}}} - V_{{N{\text{B}}}} } \right) \\ \end{aligned} $$

Equation (3a) can be simplified as

$$ U = U_{1} + U_{2} + \cdots + U_{N} $$
(3b)

Equation (3b) indicates that total voltage between A and B equals to the algebraic addition of the individual voltages.

Deduction of Formula

Thoracic Impedance Change Caused by Single Blood Vessel

As shown in Fig. 3, it is supposed that thorax is an elliptic cylindrical volume conductor with uniform conductivity, and its length is L b. BV represents a blood vessel which is parallel to the axis of thorax. The length of the blood vessel appearing between two current electrodes is equal to that of the thorax (L b), and it does not change with the expansion of the blood vessel. The real and dotted lines express its basal and expanded states, respectively. The S b0 and S b1 represent the cross-sectional areas of the basal and expanded states, respectively. ρ b is the blood resistivity. On the basis of resistance law, the impedances Z b0 and Z b1 of the basal and expanded states are expressed as Z b0 = ρ b L b/S b0 and Z b1 = ρ b L b/S b1 respectively. From the expressions of Z b0 and Z b1, we can obtain the impedance change ΔZ b caused by the expansion of the blood vessel,

Figure 3
figure 3

Single blood vessel

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$$ \begin{aligned} \Updelta Z_{\text{b}} & = Z_{{{\text{b}}1}} - Z_{{{\text{b}}0}} = {\frac{{\rho_{\text{b}} L_{\text{b}} }}{{S_{{{\text{b}}1}} }}} - {\frac{{\rho_{\text{b}} L_{\text{b}} }}{{S_{{{\text{b}}0}} }}} \\ & = \rho_{\text{b}} L_{\text{b}} \left( {{\frac{{S_{{{\text{b}}0}} - S_{{{\text{b}}1}} }}{{S_{{{\text{b}}0}} S_{{{\text{b}}1}} }}}} \right) = - {\frac{{\rho_{\text{b}} L_{\text{b}} \Updelta S_{\text{b}} }}{{S_{{{\text{b}}0}} S_{{{\text{b}}1}} }}} \\ \end{aligned} $$
(4)

In which ΔS b is the cross-sectional area change of the blood vessel (ΔS b = S b1 − S b0). When the blood vessel expands, its sectional area increases (ΔS b > 0) and the impedance decreases (ΔZ b < 0) simultaneously. Thus the minus sign is added in Eq. (4).

If the I 0 is the alternating current injecting into the human body and the S 0 is the cross-sectional area of the thorax, then the average current density j 0 in the thoracic cavity can be expressed as

$$ j_{0} = {\frac{{I_{0} }}{{S_{0} }}} $$
(5)

It is assumed that the j b is the current density of the blood and the ρ 0 is the average resistivity of the thoracic tissues. It is known from Fig. 3 that the length of the blood vessel is equal to that of thorax. When the alternating current flows through thoracic cavity (the j b and j 0 are parallel to the axis of the thorax), the alternating voltage between two endpoints of the blood vessel is equal to that of the thorax. Because the electrical field intensities E in the blood vessel and thoracic cavity are all equal to the ratio of the voltage to the length under the condition of Fig. 3, the E in the blood vessel is equal to that in thoracic cavity. According to the differential form (j = E/ρ) of Ohm’s law, the j b and j 0 can be expressed as j b = E/ρ b and j 0 = E/ρ 0, respectively, and thus E = j b ρ b and E = j 0 ρ 0. From these we obtain

$$ j_{\text{b}} = {\frac{{\rho_{0} j_{0} }}{{\rho_{\text{b}} }}} $$
(6)

On the basis of Eq. (6), when the blood vessel expands, the current I b1 through it can be expressed as

$$ I_{{{\text{b}}1}} = S_{{{\text{b}}1}} j_{\text{b}} = {\frac{{\left( {S_{{{\text{b}}0}} + \Updelta S_{\text{b}} } \right)\rho_{0} j_{0} }}{{\rho_{\text{b}} }}} $$
(7)

where S b1 = S b0 + ΔS b. In terms of Ohm’s law, the voltage change ΔU b between two endpoints (B and V) of the blood vessel can be written as ΔU b = ΔZ b I b1. Substituting Eqs. (4) and (7) into the expression for ΔU b gives

$$ \Updelta U_{\text{b}} = - {\frac{{\rho_{0} j_{0} L_{\text{b}} }}{{S_{{{\text{b}}0}} S_{{{\text{b}}1}} }}}\left( {S_{{{\text{b}}0}} \Updelta S_{\text{b}} + \Updelta S_{\text{b}} \Updelta S_{\text{b}} } \right) $$
(8)

In general, the sectional area change ΔS b of the blood vessel is smaller (S b0 ≫ ΔS b), so that the high order small item ΔS bΔS b in Eq. (8) may be neglected and Eq. (8) can proximately be simplified to

$$ \Updelta U_{\text{b}} \approx - {\frac{{\rho_{0} j_{0} L_{\text{b}} }}{{S_{{{\text{b}}1}} }}}\Updelta S_{\text{b}} $$
(9)

From Eq. (4), ΔS b = −ΔZ b S b1 S b0/ρ b L b, substituting it into Eq. (9), and rearranging it yields

$$ \Updelta U_{\text{b}} = \rho_{0} j_{0} \Updelta Z_{\text{b}} L_{\text{b}} \cdot {\frac{{S_{{{\text{b}}0}} }}{{\rho_{\text{b}} L_{\text{b}} }}} $$
(10)

According to resistance law, S b0/ρ b L b = 1/Z b0, substituting it into Eq. (10) obtains

$$ \Updelta U_{\text{b}} = {\frac{{\rho_{0} j_{0} \Updelta Z_{\text{b}} L_{\text{b}} }}{{Z_{{{\text{b}}0}} }}} $$
(11)

From Eq. (5), j 0 = I 0/S 0, substituting it into Eq. (11) gives

$$ \Updelta U_{\text{b}} = I_{0} {\frac{{\Updelta Z_{\text{b}} L_{\text{b}} \rho_{0} }}{{Z_{{{\text{b}}0}} S_{0} }}} $$
(12)

In Fig. 3, the impedance change ΔZ b is evoked when the blood vessel expands. The voltage change ΔU b resulting from ΔZ b is applied to two endpoints of the blood vessel. The electrical field distribution created by ΔU b in the thorax of the human body is similar to that in Fig. 2, so that Eq. (2) can also express the voltage change ΔU bs between the electrodes E 1 and E 2 in Fig. 3. Replacing U i in Eq. (2) with the right of Eq. (12), replacing k with k b and replacing U o with ΔU bs, and rearranging it yields

$$ \Updelta U_{\text{bs}} = k_{\text{b}} I_{0} {\frac{{\Updelta Z_{\text{b}} L_{\text{b}} }}{{Z_{{{\text{b}}0}} d}}} \cdot {\frac{{\rho_{0} L}}{{S_{0} }}} $$
(13)

where ρ 0 L/S s0 = Z s0 (the basal impedance between E 1 and E 2 in Fig. 3), substituting it into Eq. (13) gives

$$ \Updelta U_{\text{bs}} = k_{\text{b}} I_{0} {\frac{{\Updelta Z_{\text{b}} L_{\text{b}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}0}} d}}} $$
(14)

In terms of Eq. (1), the voltage change ΔU bs can be expressed as ΔU bs = I 0ΔZ bs. Substituting it into Eq. (14) and eliminating I 0, thoracic impedance change ΔZ bs caused by single blood vessel can be obtained,

$$ \Updelta Z_{\text{bs}} = k_{\text{b}} {\frac{{\Updelta Z_{\text{b}} L_{\text{b}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}0}} d}}} $$
(15a)

Rearranging Eq. (15a) for ΔZ bs/Z s0 (the ratio of the impedance change to the basal impedance on the chest surface),

$$ {\frac{{\Updelta Z_{\text{bs}} }}{{Z_{{{\text{s}}0}} }}} = k_{\text{b}} {\frac{{\Updelta Z_{\text{b}} L_{\text{b}} }}{{Z_{{{\text{b}}0}} d}}} $$
(15b)

Here k b is the proportional coefficient of a certain blood vessel. The Eqs. (15a) and (15b) signify that thoracic impedance change ΔZ bs caused by single blood vessel is directly proportional to the ratio ΔZ b/Z b0 of the impedance change to the basal impedance of the blood vessel itself, to the length L b of the blood vessel appearing between the current electrodes, and to the basal impedance Z s0 between two detective electrodes on the chest surface, while it is inversely proportional to the distance d between the blood vessel and line joining two detective electrodes.

In terms of Nyboer’s equation,10 for the above single blood vessel, we have

$$ \Updelta V_{\text{b}} = - {\frac{{\rho_{\text{b}} L_{\text{b}}^{2} \Updelta Z_{\text{b}} }}{{Z_{{{\text{b}}0}}^{2} }}} = - {\frac{{\rho_{\text{b}} L_{\text{b}}^{2} }}{{Z_{{{\text{b}}0}} }}} \cdot {\frac{{\Updelta Z_{\text{b}} }}{{Z_{{{\text{b}}0}} }}} $$
(16)

where ΔV b is the volume change of the blood vessel. As mentioned the above, Z b0 = ρ b L b/S b0, substituting it into Eq. (16) gives

$$ \Updelta V_{\text{b}} = - L_{\text{b}} S_{{{\text{b}}0}} \cdot {\frac{{\Updelta Z_{\text{b}} }}{{Z_{{{\text{b}}0}} }}} $$
(17)

where L b S b0 = V b0 (the basal volume of the blood vessel), substituting it into Eq. (17) and rearranging

$$ {\frac{{\Updelta Z_{\text{b}} }}{{Z_{{{\text{b}}0}} }}} = - {\frac{{\Updelta V_{\text{b}} }}{{V_{{{\text{b}}0}} }}} $$
(18)

Substituting Eq. (18) into Eqs. (15a) and (15b), respectively,

$$ \Updelta Z_{\text{bs}} = - k_{\text{b}} {\frac{{\Updelta V_{\text{b}} L_{\text{b}} Z_{{{\text{s}}0}} }}{{V_{{{\text{b}}0}} d}}} $$
(19a)
$$ {\frac{{\Updelta Z_{\text{bs}} }}{{Z_{{{\text{s}}0}} }}} = - k_{\text{b}} {\frac{{\Updelta V_{\text{b}} L_{\text{b}} }}{{V_{{{\text{b}}0}} d}}} $$
(19b)

where ΔV b = L bΔS b, V b0 = L b S b0, so ΔV b/V b0 = ΔS b/S b0. Substituting it into Eqs. (19a) and (19b), respectively,

$$ \Updelta Z_{\text{bs}} = - k_{\text{b}} {\frac{{\Updelta S_{\text{b}} L_{\text{b}} Z_{{{\text{s}}0}} }}{{S_{{{\text{b}}0}} d}}} $$
(20a)
$$ {\frac{{\Updelta Z_{\text{bs}} }}{{Z_{{{\text{s}}0}} }}} = - k_{\text{b}} {\frac{{\Updelta S_{\text{b}} L_{\text{b}} }}{{S_{{{\text{b}}0}} d}}} $$
(20b)

The Eqs. (19) and (20) signify that the thoracic impedance change ΔZ bs is also directly proportional to the ratio ΔV b/V b0 of the volume change to the basal volume (or the ratio ΔS b/S b0 of the sectional area change to the basal sectional area) of the blood vessel. The minus signs means that when the volume (or sectional area) of the blood vessel increases, the impedance decreases simultaneously.

Thoracic Impedance Change Caused by Multiple Blood Vessels Together

As shown in Fig. 4, it is assumed that there are multiple blood vessels (BV1, BV2,…,BV N ) in thorax, and the ΔU bs1, ΔU bs2,…,ΔU bsN are the voltage changes between E 1 and E 2 produced by their impedance changes, respectively. These voltage changes should also obey Eq. (3b), i.e. total voltage change ΔU s between E 1 and E 2 is equal to their algebraic addition,

Figure 4
figure 4

Multiple blood vessels

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$$ \Updelta U_{\text{s}} = \Updelta U_{{{\text{bs}}1}} + \Updelta U_{{{\text{bs}}2}} + \cdots + \Updelta U_{{{\text{bs}}N}} $$
(21)

Equation (21) is the specific application of Eq. (3b) in the measurement of Impedance cardiography.

It is supposed that the ΔZ s is total impedance change between E 1 and E 2, and the ΔZ bs1, ΔZ bs2,…,ΔZ bsN are the impedance changes between E 1 and E 2 caused by the individual blood vessels, respectively. According to Eq. (1), the ΔU s,ΔU bs1,ΔU bs2,…,ΔU bsN in Eq. (21) can be expressed as, respectively

$$ \begin{aligned} \Updelta U_{\text{s}} & = I_{0} \Updelta Z_{\text{s}} ,\;\Updelta U_{{{\text{bs}}1}} = I_{0} \Updelta Z_{{{\text{bs}}1}} , \\ \Updelta U_{{{\text{bs}}2}} & = I_{0} \Updelta Z_{{{\text{bs}}2}} \ldots \Updelta U_{{{\text{bs}}N}} = I_{0} \Updelta Z_{{{\text{bs}}N}} \\ \end{aligned} $$

Substituting them into Eq. (21) and eliminating I 0, thoracic impedance change ΔZ s caused by multiple blood vessels together can be obtained,

$$ \Updelta Z_{\text{s}} = \Updelta Z_{{{\text{bs}}1}} + \Updelta Z_{{{\text{bs}}2}} + \cdots + \Updelta Z_{{{\text{bs}}N}} $$
(22)

Equation (22) signifies that the thoracic impedance change ΔZ s caused by multiple blood vessels together is equal to the algebraic addition of all thoracic impedance changes (ΔZ bs1, ΔZ bs2,…,ΔZ bsN ) resulting from the individual blood vessels.

In terms of Eq. (15a), we can gain the expressions of thoracic impedance changes (ΔZ bs1, ΔZ bs2,…,ΔZ bsN ) resulting from the individual blood vessels, respectively. Substituting them into Eq. (22) gives

$$ \begin{aligned} \Updelta Z_{\text{s}} & = k_{{{\text{b}}1}} {\frac{{\Updelta Z_{{{\text{b}}1}} L_{{{\text{b}}1}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}01}} d_{1} }}} + k_{{{\text{b}}2}} {\frac{{\Updelta Z_{{{\text{b}}2}} L_{{{\text{b}}2}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}02}} d_{2} }}} + \cdots \\ & \quad + k_{{{\text{b}}N}} {\frac{{\Updelta Z_{{{\text{b}}N}} L_{{{\text{b}}N}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}0N}} d_{N} }}} \\ & = \sum\limits_{n = 1}^{N} {k_{{{\text{b}}n}} } {\frac{{\Updelta Z_{{{\text{b}}n}} L_{{{\text{b}}n}} Z_{{{\text{s}}0}} }}{{Z_{{{\text{b}}0n}} d_{n} }}} \\ \end{aligned} $$
(23)

In which N is the number of the blood vessels. Equation (23) is the concrete form of Eq. (22).

The above deduced results need to be further verified by the experiment. Because it is very difficult to measure ΔZ b, Z b0, ΔS b, S b0, L b, d in the animal or human body, we design a model experiment and validate the Eqs. (20) and (22) in detail.

Experimental Equipment

The experimental equipment used in this study is as shown in Fig. 5. In which the pane ABCD expresses the plastic barrel filling with dilute NaCl solution to simulate thorax of the human body. The inner diameter of the barrel is 38.5 cm. The height of the solution in the barrel is 45.0 cm. The detective electrodes E 1 and E 2 (2.0 × 3.0 cm2) are placed on the inner lateral wall of the barrel for detecting the impedance. The distance between the inner edges of the electrodes E 1 and E 2 is 22.0 cm.

Figure 5
figure 5

Experimental equipment

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The horizontal thick black lines in Fig. 5 represent the current electrodes I 1 and I 2 immersed in the salt solution, and both electrodes are made of the copper disks with the diameter of 38.0 cm and the thickness of 1.0 mm. The M and N represent the hooks used to hang the copper rods. The dotted line in Fig. 5 denotes a copper beam used for supporting the current electrode I 1 as well as the hooks M and N. The electrode I 1 is situated on the beam, while the I 2 is placed at the bottom of the barrel. The distance between the electrodes I 1 and E 1 as well as that between the I 2 and E 2 both are about 10 cm. The distance from the copper beam to the bottom of the barrel is about 42 cm. The distances from the hooks M and N to the straight line joining E 1 and E 2 are 20.0 and 10.0 cm, respectively (see Fig. 5).

As shown in Fig. 5, the current electrodes I 1 and I 2 are connected with the output ends of the alternating current source. The detective electrodes E 1 and E 2 are connected with the input ends of the impedance detective circuit. Its output voltage U, which is directly proportional to the impedance Z between E 1 and E 2, is sent to the computer. The instrument used to measure the impedance Z between E 1 and E 2 is a common impedance cardiograph with the computer. The root-mean-square current of its alternating current source is 1.0 mA, and the frequency is 51.2 kHz.

The slender and fat copper rods simulate the basal states and expanded states of the blood vessel, respectively. Their specifications are as shown in Table 1. In which the rods LB (1.673 cm2) and SB (2 × 1.673 cm2) simulate the basal states of the blood vessel (named the basal rod). And the rods LX1 (2.503 cm2), LX2 (3.126 cm2) and SX1 (2 × 2.503 cm2) simulate the expanded states (named the expanded rod). The LB, LX1 and LX2 are the long rods (20 cm), the SB and SX1 are the short rods (10 cm).

Table 1 Specifications of basal and expanded rods
Full size table

The impedance Z between E 1 and E 2 should be constant when the copper rods in the salt solution are not moved (see Fig. 5). However, there are the adventitious errors in the measurement. In the present study, the impedance Z between E 1 and E 2 is measured by impedance cardiograph with a sampling rate of 100 Hz over a period of 7.5 s. An average of all 750 samples is taken as the impedance Z to avoid the random measurement errors.

For the purpose of decreasing the measured errors, the equipment should be pre-heated for 0.5 h, and the solution in the barrel must be uniform. There are not any foreign substances in the solution except for the electrodes, beam, hangers and copper rods. The middle points of the copper rods and that of the line joining E 1 and E 2 are located in the same level approximately. The position and direction of the copper rods and disk current electrodes should keep to the identical condition in every experiment. Care is taken to avoid air bubble adherence to the rods.

Experimental Methods

The influences of the relative factors upon the thoracic impedance change are investigated with the experimental equipment as shown in Fig. 5. The procedures of the experiment are arranged as follows.

Influence of the Ratio (ΔS b/S b0) of the Sectional Area Change (ΔS b) to the Basal Sectional Area (S b0) of the Blood Vessel

The barrel is filled with water. NaCl (5.0 g) is added into water. The salt solution is churned to achieve the uniform condition. Then the impedance Z between E 1 and E 2 is measured when the copper rods hanged on the hooks M and N both are the basal rods LB. Its value should be in the range of 28–32 Ω (named the normal condition, NC). If Z is smaller than 28 Ω, the concentration of the solution is adjusted by means of adding water into the barrel. If Z is bigger than 32 Ω, a little NaCl is added into the solution. It must be stressed that after adding NaCl or water, the solution must be churned again, and the height of the solution should be adjusted to the level of 45.0 cm. After completing the preparation, three measurements are made:

  1. A.

    The impedance Z s0 between E 1 and E 2 is measured when the copper rods hanged on the hooks M and N both are the basal rods LB.

  2. B.

    The impedance Z 1 between E 1 and E 2 is measured when the rod hanged on the hook N is the expanded rod LX1 and that on the M is still the basal rods LB. The impedance change (ΔZ bs1 = Z 1 − Z s0) caused by LX1 is calculated.

  3. C.

    The impedance Z 2 between E 1 and E 2 is measured when the rod hanged on the hook N is another expanded rod LX2 and that on the M is still LB. The impedance change (ΔZ bs2 = Z 2 − Z s0) caused by LX2 is calculated.

The above measurements repeat for four times to decrease the measured errors. The measured data are shown in Table 2.

Table 2 Influence of the area ratio ΔS b/S b0 upon thoracic impedance change (S b0 = 1.673 cm2)
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Influence of Length L b of the Blood Vessel

  1. (1)

    Experiment with the long rods: The experiment is performed under the condition of the salt solution (NC) of part 1. Two measurements are made:

    1. A.

      The impedance Z s01 between E 1 and E 2 is measured when the rods hanged on the hooks M and N both are the long basal rods LB.

    2. B.

      The impedance Z 1 between E 1 and E 2 is measured when the rod hanged on the hook N is the long expanded rod LX1 and that on the M is still LB. The impedance change (ΔZ bs1 = Z 1 − Z s01) caused by LX1 is calculated.

The above measurements repeat for four times. The measured data are shown in Table 3.

Table 3 Influence of the vessel length L b upon thoracic impedance change
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  1. (2)

    Experiment with the short rods:

    1. A.

      The impedance Z s02 between E 1 and E 2 is measured when the rod hanged on the hook N is the short basal rod SB and that on the M is still the long basal rod LB.

    2. B.

      The impedance Z 2 between E 1 and E 2 is measured when the rod hanged on the hook N is the short expanded rod SX1 and that on the M is still LB. The impedance change (ΔZ bs2 = Z 2 − Z s02) caused by SX1 is calculated.

The above measurements repeat for four times. The measured data are shown in Table 3.

Influence of Distance d Between the Blood Vessel and Line Joining Two Detective Electrodes

The experiment is performed under the condition of the solution (NC) of part 1. Four measurements are made:

  1. A.

    The impedance Z s0 between E 1 and E 2 is measured when the rods hanged on the hooks M and N both are the basal rods LB.

  2. B.

    The impedance Z 1 between E 1 and E 2 is measured when the rod hanged on the hook N is the expanded rod LX1 and that on the M is still LB. The impedance change (ΔZ bsN = Z 1 − Z s0) is calculated.

  3. C.

    The impedance Z 2 between E 1 and E 2 is measured when the rod hanged on the hook N is the basal rod LB and that on the M is the expanded rod LX1. The impedance change (ΔZ bsM = Z 2 − Z s0) is calculated.

  4. D.

    The impedance Z 3 between E 1 and E 2 is measured when the rods hanged on the hooks M and N both are the expanded rods LX1. The impedance change (ΔZ bsMN = Z 3 − Z s0) is calculated.

The above measurements repeat for four times. The measured data are shown in Table 4.

Table 4 Influence of the distance d upon thoracic impedance change
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Influence of the Basal Impedance Z s0 Between Two Detective Electrodes on Chest Surface

  1. (1)

    Experiment with the big basal impedance: Under the condition that the rods hanged on the hooks M and N both are the basal rods LB, the impedance Z between E 1 and E 2 is adjusted to the range of 36–40 Ω through adding water into the above solution (NC) and churning it to be uniform. Two measurements are made:

    1. A.

      The impedance Z s01 between E 1 and E 2 is measured when the rods hanged on the hooks M and N both are the basal rods LB.

    2. B.

      The impedance Z 1 between E 1 and E 2 is measured when the rod hanged on the hook N is the expanded rod LX1 and that on the M is still LB. The impedance change (ΔZ bs1 = Z 1 − Z s01) is calculated.

The above measurements repeat for four times. The measured data are shown in Table 5.

Table 5 Influence of the basal impedance Z s0 upon thoracic impedance change
Full size table
  1. (2)

    Experiment with the small basal impedance: Under the condition that the rods hanged on the hooks M and N both are the basal rods LB, the impedance Z between E 1 and E 2 is adjusted to the range of 18–22 Ω through adding NaCl (5.0 g) into the above solution with the big basal impedance and churning it to be uniform. Then two measurements are made:

    1. A.

      The impedance Z s02 between E 1 and E 2 is measured when the rods hanged on the hooks M and N both are the basal rods LB.

    2. B.

      The impedance Z 2 between E 1 and E 2 is measured when the rod hanged on the hook N is the expanded rod LX1 and that on the M is still LB. The impedance change (ΔZ bs2 = Z 2 − Z s02) is calculated.

The above measurements repeat for four times. The measured data are shown in Table 5.

Result

According to the measured data in Tables 2–5, the relations between the thoracic impedance change and relative factors are analyzed.

Relation Between Thoracic Impedance Change ΔZ bs and Ratio ΔS b/S b0 of the Sectional Area Change to the Basal Sectional Area of the Blood Vessel

As shown in Table 2, the average of ΔZ bs1 caused by LX1 is −0.40 Ω, and that of ΔZ bs2 caused by LX2 is −0.68 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{{{\text{bs}}2}} }}{{\Updelta Z_{{{\text{bs}}1}} }}} = {\frac{ - 0.68}{ - 0.40}} = 1.70 $$
(24)

As shown in Table 2, the sectional area of the basal rod LB is S b0 = 1.673 cm2, the sectional area change of the expanded rod LX1 to LB is ΔS b1 = 0.830 cm2, and that of LX2 to LB is ΔS b2 = 1.453 cm2. From these we obtain

$$ {\frac{{\Updelta S_{{{\text{b}}2}} /S_{{{\text{b}}0}} }}{{\Updelta S_{{{\text{b}}1}} /S_{{{\text{b}}0}} }}} = {\frac{1.453/1.673}{0.830/1.673}} = 1.69 $$
(25a)

Since ΔV b/V b0 = L bΔS b/L b S b0 = ΔS b/S b0,

$$ {\frac{{\Updelta V_{{{\text{b}}2}} /V_{{{\text{b}}0}} }}{{\Updelta V_{{{\text{b}}1}} /V_{{{\text{b}}0}} }}} = {\frac{{\Updelta S_{{{\text{b}}2}} /S_{{{\text{b}}0}} }}{{\Updelta S_{{{\text{b}}1}} /S_{{{\text{b}}0}} }}} = 1.69 $$
(25b)

The ratios (1.69) in the expressions (25a) and (25b) are in accord with the ratio (1.70) in the expression (24) basically. This indicates that the thoracic impedance change ΔZ bs is directly proportional to the ratio ΔS b/S b0 of the sectional area change to basal sectional area (or the ratio ΔV b/V b0 of the volume change to basal volume) of the blood vessel.

Relation Between Thoracic Impedance Change ΔZ bs and Length L b of the Blood Vessel

As shown in Table 3, in the experiment with the long rods the average of ΔZ bs1 is −0.40 Ω, and that of Z s01 is 29.33 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{\text{bs1}} }}{{Z_{{{\text{s0}}1}} }}} = {\frac{ - 0.40}{29.33}} = - 0.0136 $$
(26)

In the experiment with the short rods, the average of ΔZ bs2 is −0.21 Ω, and that of Z s02 is 31.70 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{\text{bs2}} }}{{Z_{\text{s02}} }}} = {\frac{ - 0.21}{31.70}} = - 0.0066 $$
(27)

From two ratios in the expressions (26) and (27), we obtain −0.0066/−0.0136 = 0.485. While the longitudinal ratio of the short rod to long rod is 10.0/20.0 = 0.50. It is in accord with the above ratio (0.485) basically. This indicates that thoracic impedance change ΔZ bs is directly proportional to the length L b of the blood vessel.

Relation Between Thoracic Impedance Change ΔZ bs and the Distance d Between the Blood Vessel and line Joining Two Detective Electrodes

As shown in Table 4, the average of ΔZ bsN is −0.41 Ω, and that of ΔZ bsM is −0.21 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{\text{bsN}} }}{{\Updelta Z_{\text{bsM}} }}} = {\frac{ - 0.41}{ - 0.21}} = 1.95 $$
(28)

As shown in Fig. 5, the distances from the hangers N and M to the line joining E 1 and E 2 are d N = 10.0 cm and d M = 20.0 cm, respectively, their ratio is

$$ {\frac{{d_{\text{N}} }}{{d_{\text{M}} }}} = {\frac{10.0}{20.0}} = 0.50 $$
(29)

The ratio (1.95) in the expression (28) is equal to the reciprocal of the ratio (0.50) in the expression (29) approximately, and vice versa. This indicates that thoracic impedance change ΔZ bs is inversely proportional to the distance d from the blood vessel to the line joining two detective electrodes.

Relation Between Thoracic Impedance Change ΔZ bs and Basal Impedance Z s0 Between Two Detective Electrodes on Chest Surface

As shown in Table 5, in case of the big basal impedance the average of ΔZ bs1 is −0.50 Ω, and that of Z s01 is 37.52 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{\text{bs1}} }}{{Z_{{{\text{s0}}1}} }}} = {\frac{ - 0.50}{37.52}} = - 0.0133 $$
(30)

In case of the small basal impedance, the average of ΔZ bs2 is −0.27 Ω, and that of Z s02 is 19.90 Ω. Their ratio is

$$ {\frac{{\Updelta Z_{\text{bs2}} }}{{Z_{\text{s02}} }}} = {\frac{ - 0.27}{19.90}} = - 0.0136 $$
(31)

The ratio (−0.0136) in the expression (31) is in accord with the ratio (−0.0133) in the expression (30) basically. This indicates that thoracic impedance change ΔZ bs is directly proportional to the basal impedance Z s0 between two detective electrodes on the chest surface.

Addition of Impedance Changes

As shown in Table 4, when the basal rod LB hanged on the hanger N is alone replaced by the expanded rod LX1, the average of the impedance change ΔZ bsN is −0.41 Ω. When the basal rod LB hanged on the hanger M is alone replaced by the expanded rod LX1, the average of ΔZ bsM is −0.21 Ω. The algebraic addition of two impedance changes is ΔZ bsN + ΔZ bsM = −0.62 Ω. When the basal rods LB hanged on the hangers N and M both are replaced by the expanded rods LX1, the average of the impedance change ΔZ bsMN is −0.61 Ω. It is in accord with ΔZ bsN + ΔZ bsM basically. This indicates that thoracic impedance change caused by multiple blood vessels together is equal to the algebraic addition of all thoracic impedance changes resulting from the individual blood vessels.

Discussion

From the above derivation process it is known that when the alternating current I 0 flows through thorax of the human body, the impedance change ΔZ b of the blood vessel can cause the voltage change ΔU b between its two endpoints. The ΔU b can induce the electrical field in thorax. The detective electrodes E 1 and E 2 on the chest surface are two points in this electrical field, and they are not located on a same equipotential line.

According to the electrical theory,9 there is a potential difference (i.e. the voltage change ΔU bs) between E 1 and E 2, and ΔU bs should correspond to an impedance change on the chest surface. It is just thoracic impedance change ΔZ bs, and ΔU bs = I 0ΔZ bs. Because the root-mean-square current of I 0 is constant, the ΔU bs is directly proportional to ΔZ bs. Thus the ΔZ bs can be obtained by means of measuring the voltage change ΔU bs between E 1 and E 2 on the chest surface. On the basis of Eqs. (15a) and (18), the ΔZ bs is directly proportional to ΔZ b (note that ΔZ bs ≪ ΔZ b), and the ΔZ b is directly proportional to the volume change ΔV b of the blood vessel. Therefore, the voltage change ΔU bs measured on the chest surface can indirectly reflect the impedance change and the volume change of the blood vessel in thoracic cavity.

Synthesizing the above experimental results, it can be shown that thoracic impedance change caused by single blood vessel is directly proportional to the ratio of the sectional area change to the basal sectional area of the blood vessel itself, to the length of the blood vessel appearing between the current electrodes, and to the basal impedance between two detective electrodes on the chest surface, while it is inversely proportional to the distance between the blood vessel and line joining two detective electrodes. The thoracic impedance change caused by multiple blood vessels together is equal to the algebraic addition of all thoracic impedance changes resulting from the individual blood vessels. That is, the impedance changes obey the principle of adding scalars in the measurement of the electrical impedance graph. Thus Eqs. (20) and (22) are confirmed by the model experiment. It is also suggested that the theoretical model and deduced method in this paper are feasible.

In the above experiments, in order to decrease the measured errors of the impedance change ΔZ bs, the volume changes of the expanded rods against the basal rods are bigger (ΔV b1/V b0 = 49.6% for LX1, ΔV b2/V b0 = 86.8% for LX2). Under this condition, the relationship between the impedance change and the volume change does not obey Nyboer’s equation.10 On the basis of Eq. (4), ΔZ b1 = ρ b L bΔS b1/S b0 S b1 for LX1, ΔZ b2 = ρ b L bΔS b2/S b0 S b2 for LX2. From these we obtain

$$ {\frac{{\Updelta Z_{{{\text{b}}2}} /Z_{{{\text{b}}0}} }}{{\Updelta Z_{{{\text{b}}1}} /Z_{{{\text{b}}0}} }}} = {\frac{{\Updelta S_{{{\text{b}}2}} \cdot S_{\text{b1}} }}{{\Updelta S_{{{\text{b}}1}} \cdot S_{\text{b2}} }}} $$
(32)

From Tables 1 and 2, ΔS b1 = 0.830 cm2, ΔS b2 = 1.453 cm2, S b1 = 2.503 cm2, S b2 = 3.126 cm2. Substituting them into the expression (32) gives

$$ {\frac{{\Updelta Z_{{{\text{b}}2}} /Z_{{{\text{b}}0}} }}{{\Updelta Z_{{{\text{b}}1}} /Z_{{{\text{b}}0}} }}} = {\frac{1.453 \times 2.503}{0.830 \times 3.126}} = 1.40 $$
(33)

It manifests that under the above experimental condition, the difference between the ratio (1.40) in the expression (33) and the ratio (1.70) in the expression (24) is bigger, and ΔZ bs is not directly proportional to the ratio ΔZ b/Z b0. But in the human body, the volume changes of the blood vessels are usually smaller. Equation (18) is proximately tenable, and the direct proportional relation between ΔZ bs and ΔZ b/Z b0 is still kept.

In the experiment of the influence of the length L b upon the impedance change ΔZ bs, the volume of the short basal rod SB equals to that of the long basal rod LB (see Table 1), the volume of the short expanded rod SX1 equals to that of the long expanded rod LX1, and the volume change of SX1 against SB is equal to that of LX1 against LB. In such a case, it is obtained that the impedance change ΔZ bs is directly proportional to the length L b of the blood vessel. But in the experiment, it is also found that if the sectional area of SB equals to that of LB and the sectional area of SX1 equals to that of LX1 (i.e. the volume of SB is one half of LB and the volume of SX1 is one half of LX1), the impedance change ΔZ bs is not directly proportional to the length L b of the blood vessel.

The deduced process of Eq. (22) reveals that the addition of the impedance changes involved in the measurement of the electrical impedance graph is actually the addition of the corresponding voltage changes. It corresponds to the electrical theory (Eq. 3b). In part 2 and part 4 of the above experiments, the measured data in Tables 3 and 5 are treated with the ratio ΔZ bs/Z s0 because the impedance change ΔZ bs is influenced by the basal impedance Z s0 on the chest surface.

The frequency used to measure the human impedance is currently in the range of 20–100 kHz, but the frequency used in different area is not the same.3 , 8 The frequency of 100 kHz is usually used in the United States of America, while the frequency of 50 kHz is currently used in China and Japan (the technical specification: 50 kHz ± 10%). As mentioned previously, the frequency used in the present study is 51.2 kHz, and corresponds to the above technical requirement. This frequency is not specified for the present study, but it must be very stable because the capacitive component of the impedance is changed with the frequency. For this reason, the frequency of 51.2 kHz is supplied by the crystal oscillator.

Conclusion

In this study, based on Ohm’ law and the electrical field distribution in cylindrical volume conductor, the formula about thoracic impedance change have been deduced, and the mechanism of the formation for thoracic impedance change has been explained. They have been confirmed by the model experiment. These results could be used as the theoretical basis for the waveform reconstruction of Impedance cardiography.