BE 310FINAL REPORT

DATE OF SUBMISSION: 05/06/2004

GROUP: M3

TITLE: IMPEDANCE CARDIOGRAPHY

GROUP MEMBERS :Amar Bansal

Prasheel LillaneyAnand Patel

Shaunak Patel

Abstract

Changing internal body configurations such as a contracted or relaxed heart result in body impedance changes as well. In the experiment, cardiac output was measured from subjects using the established Kubicek formulation and values of approximately 25 L/min were obtained. These largely flawed measurements in cardiac output (compared to literature values of 5 L/min) suggest that either the Kubicek formulation or measurement may be faulty. In this study, using finite element modeling with PDEase software, an alternate value for the change in body impedance was calculated (3.87 ohm/sec) by using physiological geometry by modeling a torso with a contracted and relaxed heart. It was determined that since this impedance value was approximately equal to the measured value (2.05 ohm/sec), this implies that it is Kubicek's formulation, not the impedance measurement, which is the source of error for determining cardiac output.

Introduction

Cardiac output is defined as the volume of blood pumped out by the heart in a given interval of time (L/min). The initial goal of this experiment was to measure cardiac output non-invasively. One of the methods by which this can be done is by using the body’s impedance changes. Changing internal body configurations such as a contracted or relaxed heart result in body impedance changes as well. In the experiment cardiac output was measured from subjects using the established Kubicek formulation. However, largely erroneous measurements in cardiac output suggest that either the formulation or measurement may be faulty. In this study, using finite element modeling, an alternate value for the change in body impedance was calculated using physiological geometry. It was determined that since this impedance value was approximately equal to the measured value, this implies that it is Kubicek's formulation, not the impedance measurement, which is the source of error for finding cardiac output.

Mathematical Background

In order to determine the cardiac output using body impedance, only the impedance change over the upper thoracic region is considered. Therefore, the main physiological change that would affect the impedance would be the contraction and expansion of the heart. This is because of the volume change of blood, which has a conductivity that is significantly higher than most of the surrounding tissue. Thus it is expected that the thoracic impedance reaches a minimum value when the heart ventricles are relaxed.

The first model used was the Kubicek model. This model approximates the heart and thorax to be cylinders. The heart is approximated to a cylinder with fixed height l and variable cross sectional area A. Therefore the stroke volume (SV) which is defined as the volume of blood ejected in one contraction is given by the formula:

(1)

Here dA/dt is the change in cross-sectional area with respect to time and TLVE is the time interval of the left ventricular ejection. In order to derive the Kubicek model, it is necessary to find relationships between the cross sectional area and impedance changes. The reasoning behind this is that the cross sectional area is very hard to measure non-invasively but the changes in thoracic impedance can be measured in this manner. In order to arrive at the above mentioned relationships, another approximation must be made that states that the body is a pure resistance model, i.e. there are no inductance or

capacitance terms in the impedance equation. Therefore the impedance of the body Z can be given by:

(2)

where ρ represents the resistivity.

Equation (2) can be differentiated using the chain rule to obtain the function for the impedance change with respect to time (dZ/dt):

(3)

Substituting equation (2) into equation (3) yields:

(4)

where Z0 is the base line impedance of the body.

Finally, by substituting equation (4) back into equation (1) yields:

(5)Therefore the Kubicek model obtains a relationship between cardiac output and externally measurable quantities Z and dZ/dt. The final approximation in the Kubicek model is the value of dZ/dt, for which the maximum dZ/dt value is taken.

Materials and Methods

The apparatus used in this experiment were:1. Biopac MP30 acquisition unit2. Biopac EL503 spot electrodes3. Biopac EL506 strip electrodes4. Biopac SS31L non-invasive cardiac output sensor5. Biopac plethysmograph6. Biopac SS2L, SS31L electrode leads 7. Biopac SS30L stethoscope8. rubbing alcohol9. cotton swab10. meter stick

Each subject was connected as follows. Four EL506 strip electrodes were used on each subject. Two were placed horizontally on the base of the neck approximately 3 centimeters apart, and the other two were placed on the back in the diaphragm area. The distance between the two pairs was kept constant at 27 centimeter. The electrode leads were connected as follows: white to neck (top), red to neck (bottom), green to back (top)

and black (ground) to back (bottom). These leads are connected to the Biopac SS31L sensor and are responsible for the thoracic impedance measurement.

Two EL503 spot electrodes were connected to each subject- one to the right wrist and one to the left ankle. The SS2L leads were connected as follows: white to right wrist, red to left ankle. There is no ground electrode connected. These electrodes are responsible for measuring the ECG signal. Before attachment, alcohol was applied to the area, after which the skin was rubbed dry using a cotton swab. This was done in order to improve the contact between the electrode and skin.

The Biopac plethysmograph sensor was attached to the tip of the right index finger of the subject. this sensor measured the subjects pulse. Finally, the SS30L stethoscope was placed on the left side of the chest at the base of one of the pectoral muscles. This sensor was used to positively determine the time interval of the left ventricular ejection.

Each subject was tested in three positions- lying down, sitting and standing. Data was collected for a time interval of a minute.

Figure 1: Template for data collection and Kubicek calculation of cardiac output.

All sensor readings were input to a pre-programmed template which calculated real-time dZ/dt, stroke volume and cardiac output. A modified template is shown below. Note that the dZ/dt and pulse signals are not included in this figure. The mean value of Z (the first waveform) is the base line value of body impedance Z0. The stroke volume waveform is obtained by applying equation (5) previously obtained to the dZ/dt waveform. Similarly, heart rate is obtained from the ECG data. The product of stroke volume and heart rate is the cardiac output which is shown in the bottom most waveform.

For every position of the two subjects, ten values of maximum dZ, integral of Z over TLVE

and cardiac output were obtained. These values were then compared to literature values and to values obtained from different models.

Results – Kubicek Model

Using the EL503 spot electrodes and the EL506 strip electrodes, impedance data were collected from two subjects. Subjects assumed three different positions (lying, sitting, and standing). As discussed in the mathematical background, dZ/dt values are used to determine the cardiac output according to the following equation:

Table 1 and 2 show the dZ/dt, delta Z, and cardiac output values that were obtained for the two subjects tested.

 dZ/dt Max (Ohm/s) Delta Z (Ohm)

Cardiac Output (L/min)

Lying 2.42 ± 0.38 0.26 ± 0.05 26.3 ± 3.1Sitting 1.90 ± 0.22 0.19 ± 0.05 28.7 ± 8.1Standing 1.90 ± 0.16 0.18 ± 0.02 31.2 ± 9.1Table 1: dz/dt, delta z, and cardiac output values for subject 1

 dZ/dt Max (Ohm/s) Delta Z (Ohm)

Cardiac Output (L/min)

Lying 2.38 ± 0.34 0.24 ± 0.03 23.4 ± 4.2Sitting 1.66 ± 0.20 0.23 ± 0.04 19.5 ± 7.6Standing 1.48 ± 0.22 0.18 ± 0.05 17.0 ± 6.8Table 2: dz/dt, delta z, and cardiac output values for subject 2

The values for cardiac output shown in Tables 1 and 2 are clearly erroneous because typical values are near 5 L/min. This large error can be attributed to one of two factors: a) the instrumentation used to obtain the raw data (i.e., electrodes, MP30 unit, etc) is defective or b) the Kubicek model that calculates the cardiac output from the raw data is inaccurate. The main concern with the former factor was to determine whether or not the measurement of the dZ/dt value was accurate. Hence, the goal of the second part of this study was to construct a new model that would predict what values of dZ/dt should be expected in a typical human torso.

Finite Element Study Background

In order to predict the value of dZ/dt a simulation was constructed using the finite element method. An overview of this process is given below in order to understand the scope of the procedure.

The first step in the process was to define a cylindrically symmetric cross section of the human torso (Figure 2) and ‘digitize’ the image, where the term ‘digitize’ simply means to define a finite number of points (r,z) on the boundary of the image so that it can be represented on a 2D coordinate system. The cross section was scanned in from a general anatomy text and scaled for a total body length of 2 meters. In order to model the heart as a sphere in this cylindrically symmetric cross section, a semi-circle was inserted approximately at the middle of the chest so that if the image is revolved around the z-axis the semi-circle would become a sphere.

After defining the torso a choice had to be made in respect to how the simulation was going to be run. The choices were to either apply a test voltage and to simulate the measurement a test current or to apply a test current and simulate the measurement of the corresponding test voltage. In either case the test voltage divided by the test current would give the magnitude of the impedance between the terminals where the current or voltage was applied. In this simulation, the finite element model solves Laplace’s equation (which will be discussed further below) for the voltage potential in the torso. Thus, the majority of the boundary conditions would be in terms of the voltage potential, therefore, the choice was made to apply a test voltage instead of a test current. For simplicity, a 1 V clamp was placed at the neck, and the waist was clamped at ground (note: these clamps are represented by the red lines in Figure 2).

In order to understand how a value for dZ/dt will be calculated, the mathematics of the system must be discussed further. As mentioned above, Laplace’s Equation was solved for the voltage potential in the torso. Due to the fact that the body was being separated into different regions (the heart and the remainder of the body), the conductivity (or conversely the resistivity) must be a function of the special coordinates of the system. Hence the form of Laplace’s equation, which holds for any orthogonal coordinate system, that applied to the problem is shown below. Here, V represents the voltage potential as a function of r and z, while sigma is the spatially varying conductivity (where the

Figure 2 – Cylindrically symmetric cross section for human torso used for the finite element study. The circular region in the center of the chest is meant to represent the heart. Only half of the image was needed since the system is defined to be cylindrically symmetric but the whole image is shown here for clarity.

r

z

conductivity of the heart region was 1 S-m while the conductivity of the rest of the body was 0.2 S-m):

The boundary conditions for the problem were both Dirichlet and Neumann. The Dirichlet boundary conditions occurred at the voltage clamps since the voltage was fixed at 1 V at the neck and 0 V (ground) at the waist. The Neumann condition occurred around the whole boundary of the torso because it was assumed that no current leaked through the boundary of the body. As a result the normal derivative of the voltage potential at the boundary would have to be 0. Lastly, it was assumed that there was no charge build up in the heart region.

The above mathematical development only provides a solution for the voltage potential in the torso. However, in order to simulate the measurement of the magnitude of the impedance, the net current flowing through the torso must also be solved for. In order to do this the definition of current density must be introduced, and is shown below (here the current density is defined in the z direction):

In order to determine the net current flowing in the z direction the current density must be integrated as shown below where the upper limit R in the second integral simply represents the general edge of the torso:

An important aspect of this integral is that regardless of what height z is chosen to integrate along in the r direction (as long as the z chosen is in between the z values of the upper and lower voltage clamps), the value of the integral will be constant because current is conserved since there is no leakage out of the body and there is no build up in the heart. Once the current is calculated the magnitude of the impedance is simply equal to 1 V / current.

Up to this point the variable of time still has not been introduced into the system, which may seem strange because the overall goal is to arrive at a value for dZ/dt. In order to approximate this value of dZ/dt and not formally introduce the variable of time into the system, the finite element model was solved under two separate states. The first state was the 'contracted' state in which the heart had a radius of 4 cm, while the second state was the 'relaxed' state in which the heart had a radius of 6 cm. A value for the magnitude of the impedance for the thorax was calculated for each of these states and it was

assumed that the time difference between states was 0.3 seconds as this was the time used for left ventricular ejection in the Kubicek model on biopac. Thus the value of dZ/dt was approximated by (impedance 'contracted' – impedance 'relaxed') / 0.3 seconds.

Finite Element Results

The finite element program, PDEase was used in order to simulate the contracted and relaxed state and thus find a dz/dt value with the addition of the heart. In order to find a numerical solution, PDEase divided the digitized body into a mesh of finite elements (approximately 15,000 nodes) for the contracted and relaxed states (Figure 3). The software then used this nodal configuration to numerically solve Laplace’s equation for each state (Figure 4). Essentially, these solutions are contour maps of the electric potential for both the contracted and relaxed states.

Figure 3 – Nodal, finite element solutions for the contracted (left) and relaxed (right) states. 15,213 nodes were mapped for the contracted state, and 15,423 nodes were mapped for the relaxed state.

Figure 4 – Numerical solutions for the electric potential for the contracted (left) and relaxed (right) states. Note how the isopotential lines around the heart tend to curve as the heart (region of higher conductivity) expands.

Once the potential solutions for each state were determined, in order to find the current through a cross section of the body, the surface integral of current density was calculated on an arbitrary cross section from the r = 0 (the centerline) to the outer boundary of the body. As a side, with the PDEase software student version, the calculation of the surface integral required input of a ‘2πrdr’ term due to the cylindrical geometry; however, in the professional version of the software when evaluating a surface integral in cylindrical coordinates it automatically take into account for this term. As seen in Figure 5, by finding the area under the curve (current density vs r), the software determined the value of the total current to be 18.66 mA for the contracted state and 19.07 mA for the relaxed state. Since a test voltage of 1 V was applied, and since Z=V/I, this immediately implies that the impedance for the contracted state was (1/18.66mA) 53.60 ohms, while the impedance for the relaxed state was (1/19.07mA) 52.44 ohms. Next, in order to find dz/dt, the following approximation was used:

Here, Tc is the time of ventricular contraction, which was taken to be normal, average physiological value of 0.3 seconds. Thus, it was determined that dz/dt = 3.87 ohms/sec. It is noted that while approximately 15,000 nodes were used for the potential calculations, the calculations were repeated with as many as 150,000 nodes; however, the current values only differed by 0.1 mA, and therefore it is reasonable that 15,000 nodes provided sufficient accuracy. Moreover, using 15,000 nodes was easier in that it avoided the cumbersome nature of using extremely large data files.

Figure 5 – Plot of current density vs r for a given cross section shown as the marked horizontal line in the small torso figure on the right side of the image.

Discussion

By performing several tests with the SS31L sensor on Biopac it was possible to see that the average value of dZ/dt was 2.05 ohm/sec, while the FEM analysis yielded a value of 3.87 ohm/sec. Although the percent difference between these values is approximately 100%, it is important to remember that the simulated FEM dZ/dt is only an estimate of the actual measurement. Furthermore, the FEM value of dZ/dt was only meant to verify that the SS31L sensor was providing measurements in the general range of what was expected. Since the dZ/dt values from the SS31L sensor and the FEM analysis are well within the same order of magnitude it is possible to conclude that the measurement being made is indeed accurate, and the actual source of error in the erroneous cardiac output values derives from the Kubicek formula.

It is interesting to compare these results to a similar, but far more complex, FEM study carried out by Wang et al (Figure 6). In this study, not only was the effect of ventricular contraction (V) considered, but the effects of major vessel expansion (M), blood resistivity changes (B), and lung resistivity changes (L) were also examined. When looking at the effect of V on the entire thorax, the study found that the impedance change between the contracted and relaxed states was approximately 0.75 ohm or 2.5 ohm/sec, if a time period of 0.3 seconds is used between the states. This is in general agreement with the FEM value calculated in this study of 3.87 ohm/sec. However, if all of the above mentioned effects are combined (V+M+B+L), Wang's results show that the impedance change for the entire thorax between the contracted and relaxed states actually has a value of -0.5 ohm or -1.67 ohm/sec. This strongly disagrees with the measurement made by the SS31L sensor (+2.05 ohm/sec), which should measure the effect of V+M+B+L. This contradiction is elucidated when looking at Wang's results for only the upper thorax, which is a more appropriate comparison since the electrodes for the SS31L sensor were not placed at the neck and waste, but rather at the neck and mid-back. Wang's results for V+M+B+L for the upper thorax show an impedance change between contracted and relaxed of approximately 1.2 ohm or 4 ohm/sec, which is in better agreement with what the SS31L sensor actually measured (2.05 ohm/sec). Wang's results in this case may be an overestimate because they correspond to electrodes from the neck to the upper back and not the mid-back.

Overall, it is possible to see that the physical measurements made in this study do correspond well with the FEM simulation and literature results. Furthermore, the FEM analysis done in this study also corresponds to the more comprehensive FEM study completed by Wang. Thus, it is safe to conclude that the dZ/dt value is indeed measured accurately, and again the Kubicek formula for cardiac output is in this case the major source of error due to the fact that it is not very anatomically correct. However, one may point out that the Kubicek model may have some clinical relevance since it still may be used to broadly monitor cardiac output (order of magnitude accuracy), if the electrodes are placed in a manner to measure only the impedance change in the upper thorax. The reasoning behind this is that in the case of the upper thorax, the impedance change is dominated by the effect of ventricular contraction (Figure 6). Thus, the impedance change measured in the upper thorax will still correspond to ventricular contraction, but using this impedance change to actually determine an exact value for cardiac output is severely inaccurate.

Acknowledgements

We would like to thank Dr. Ken Foster for providing support and instruction with the finite element modeling, and for providing us with the fantastic software, PDEase.

References

Figure 6- Results from Wang, which demonstrate that when looking at just V for the entire thorax there is a positive impedance change between contracted and relaxed states. However, when examining the net change for the entire thorax the result is actually a negative impedance change. Opposite results are shown when examining the net results for only the upper thorax. In this case the impedance change is still a positive value.

W.G. Kubicek, J.N. Karnegis, R.P. Patterson, D.A. Witsoe, and R.H. Mattson, “Development and Evaluation of an Impedance Cardiac Output System,” Aerospace Medicine, vol. 37, pp. 1208-1212, Dec. 1966.

Y. Wang, D.R. Haynor, and Y. Kim, “A Finite-Element Study of the Effects of Electrode Position on the Measured Impedance Change in Impedance Cardiography,” IEEE Trans. Biomed. Eng., vol. 48, no. 12, pp. 1390-1401, Dec. 2001.

E. Raaijmakers, Th.J.C. Faes, H.G. Goovaerts, P.M.J.M. de Vries, and R.M. Heethaar, “The Inaccuracy of Kubicek’s One-Cylinder Model in Thoracic Impedance Cardiography,” IEEE Trans. Biomed. Eng., vol. 44, no.1, pp. 70-76, Jan.1997.