Magnetic induction pneumography: a planar coil system for continuous monitoring of lung function via contactless measurements

The forward problem is defined as, given the electrical properties of the body and the geometry of the coil system, solving for the induced voltages in the receiver coils. This requires the solution of Maxwell’s electromagnetic equations which was reported in several earlier studies (see, for example [13]). A brief review is given here to provide a background and set the stage for the inverse problem.

Assume an AC driven coil located in the proximity of a volume conductor which is represented by Ω. The forward problem is to compute the induced voltages in the remaining pick-up coils due to the eddy-currents in Ω. According to Geselowitz’s reciprocity theorem, the induced voltage can be expressed as follows,

where E₁(.) and E₂(.) represent the direct and adjoint electric fields induced in the body, respectively. E₁(.) is computed directly from the field of the driven coil and E₂(.) is computed assuming a unity current in the pick-up coil as if it acts as a transmitter coil. σ represents the electrical conductivity distribution within Ω. The computation of E₁(.) and E₂(.) requires the solution of elliptic partial differential equations and the computational details can be found in an earlier study [14].

In MIP and all related impedance imaging techniques, applying a differential measurement strategy is favorable because the systematic errors and drifts are better suppressed in this way. However, if the measurements are taken at different operating frequencies, unlike EIT, a scaling is necessary [15] as given below,

In here, v_a and v_b, respectively, represent two sets of voltages for two distinct conductivity distributions, σ_a and σ_b, acquired at angular frequencies w_a and w_b. If the data are acquired at the same frequency, as in the case of time-differential measurements, then the scaling factor (w_a/w_b)² becomes unity.

Presently linearization approaches are commonly used to deal with the nonlinearity of the forward map. By linearizing (1), the change of voltage data, ᐃv, due to a conductivity perturbation, ᐃσ, around a particular distribution, σ, can be approximated from [16],

When a piecewise constant conductivity distribution, i.e., σ = [σ₁,σ₂, · · · ,σ_N]^T , is assumed within Ω, (3) can be expressed as,

where N is the number of voxels, i.e., the number of elements in the FE mesh. By introducing S(.) as the sensitivity matrix, (4) can be expressed by a linear matrix equation which relates the perturbation in the conductivity distribution to the perturbation in the voltage data,

The entries of S are obtained by successively computing the integral in (4) for different pairs of transmit-receive coil. For instance, assuming M transmitter and N pick-up coils, S consists of M × N rows, each of which corresponds to an independent set of voltage data. It should be noted that this approximation is valid only for limited conductivity fluctuations around σ for which the linear approximation error stays below a defined threshold.

The image reconstruction problem is to restore the conductivity perturbation, Δσ^,$$\Delta \hat{\sigma },$$from the measurement data, ᐃv. By applying SVD to the sensitivity matrix, i.e., S = UΣV^T , and performing the inverse, Δσ^$$\Delta \hat{\sigma }$$can be approximated as follows,

In here, U and V are the singular vectors which form a basis for the data and the model space, respectively, and Σ^† is a diagonal matrix, the diagonal entries of which are the reciprocal of the singular values. Due to the ill-conditioning of S, which can be noted from the fast decay of singular values presented in figure 2, a truncation by using only the t column vectors of U and V corresponding to the t largest singular values is preferred to stabilize the inversion [17].

Fig. 2

Let ρ₁ be the largest singular value and ρt the tth largest one, the proper selection of the truncation level t, was performed by selecting the maximum t that satisfies the following equality:

where ᐃv_RMS and δ_RMS are the root-mean-square of the perturbed voltage signal and noise signal, respectively. The left-hand side is referred to as the SNR. When the noise is considered to be additive white Gaussian noise (AWGN) with zero mean, then, δ_RMS denotes the standard deviation of the noise signal.

Assuming that the measurement data are uncorrelated and the data variances are normalized to unity, the image resolution can be expressed in terms of the SVD components as follows [18],

Thus, the resolution matrix is simply VV^T considering only the singular vectors corresponding to the t largest singular values. Ideally, R would be an identity matrix, however, a perfect resolution can never be attained in practice due to the nonzero measurement noise. The columns of R are commonly referred to as the PSFs, which describe the disturbance in the reconstructed image due to a conductivity perturbation in a single voxel.

A 5× 5 planar coil arrangement used in simulations is depicted in figure 1. The solenoid coils were approximated as infinitely thin circular wires. They are assumed to act both as transmitters and receivers and have a radius of 2.5 cm. The sensors cover a square region of 625 cm² in total. For the simulations, the approximate distance from the coil center to the body surface was 1 cm. Figure 3 illustrates the coil geometry used in simulations with respect to the 3-D chest model.

Fig. 3

The tissue boundaries of the chest model was segmented from the NMR images to construct a three dimensional volume. Typically, reconstruction of small details is infeasible with diffused imaging modalities; therefore, the model was constructed roughly by considering only the extended tissue groups such as the lungs, heart and the surrounding muscle tissue. Two different tetrahedral meshes with different sizes of approximately 100000 and 60000 elements were generated to be used independently for voltage simulation and image reconstruction, respectively. The electrical conductivities of the tissues were taken as 0.27, 0.11, 0.21 and 0.36 Sm⁻¹ for deflated lung, inflated lung, heart and muscle tissues, respectively, corresponding to 100 kHz operating frequency [12].

The fluid accumulation that causes edematous lung injury takes usually place when the patients spend long time immobilized in supine position, and thus, gravitation forces a marked amount of fluid accumulate in the posterior parts of the lungs. The collapsed region is commonly a mixture of edematous fluid together with the deflated lung tissue causing the conductivity of that region to be higher than the healthy tissue. A conductivity value of 0.6 Sm⁻¹ was assigned to the collapsed region according to [19]. The formation of edema in the posterior regions of the lungs is illustrated as red regions in figure 3.

The sensitivity matrix of the design was calculated assuming that the coil plane is located 1 cm above a half-space that has a uniform conductivity distribution. The sensitivity distributions for different coils in the sensor matrix are presented in figure 4. Both the sensitivity distribution near the surface and on the transverse plane are shown. For all setups, the region between the coils provides positive voltages, i.e., positive sensitive regions. The black lines represent the height and width of the regions where the sensitivity drops to 95% of the maximum sensitivity. The depth sensitivity increases with the increasing relative distance between the coils. However, further increase of the distance than 15 cm did not show any improvement in depth sensitivity.

Fig. 4

The resolution matrix was computed using the same half-space model by assuming that the coil plane is located 1 cm above the half-space. To determine the truncation level necessary for a stable image reconstruction, the logarithmic ratio between the first and the remaining singular values as compared to the SNR levels was examined. The optimum truncation level for the designs corresponding to the SNR levels of 40 dB, 60 dB and 80 dB were found to be 145, 216 and 275, respectively. The PSFs of several locations for the SNR level of 60 dB are presented in figure 5. Near the surface regions, the PSFs are sharp and narrow, however, with increasing depth, they become wider and show more diffuse characteristics. Also the maximum amplitude of the PSFs approximately drops exponentially with depth. Besides, the peak of the PSFs shift towards the surface regions (more in deeper regions and less in surface-near regions) and thus, the localization errors become more prominent with increasing depth.

Fig. 5

Assume that the patient is in a supine position and the measurements are taken at different times, t₁ and t₂. Using time-differential voltage measurements, i.e., ᐃv = v(t₂) − v(t₁), one can monitor the change of electrical conductivity in time due to breathing, i.e., ᐃσ = σ(t₂) −σ(t₁). The measurements were computed at t₁ and t₂, that correspond to the deflated and inflated lungs, respectively, while the thorax geometry is fixed. 3 sets of voltage data were generated by adding AWGN that correspond to 40 dB, 60 dB and 80 dB SNR levels. For the image reconstruction, the sensitivity matrix was computed based on the deflated lung state where the tissue boundaries can be segmented from high resolution imaging modalities. Thus, the drop in conductivity due to inflation of the lungs is aimed to be imaged. The optimal truncation parameter t was found to be 130, 227 and 300 for the corresponding SNR levels of 40 dB, 60 dB and 80 dB SNR, respectively. The reconstructed images due to the temporal conductivity changes during ventilation are presented in figure 7. The slices correspond to the transverse plane of the thorax model that is orthogonal to the sensor plane. Figure 6 illustrates the original model that corresponds to that slice. The recontructed images in figure 7 show blurry characteristics, however, both lung regions can be seperately identified at the dorsal region. Strong negative artifacts were noted between the lungs, particularly for the case of 60 dB and 80 dB SNR. Several positive and negative “spiky” artifacts were noted particularly near the dorsal surface.

Fig. 6

Fig. 7

Is some cases, especially for the patients that spend long time in a supine posion, a markedly increased gravitational fluid accumulation hinders normal functioning of the lungs in the posterior regions. To simulate this, it was assumed that the fluid accumulation has reached a state in which 2.5 cm, 5 cm and 7.5 cm of the lower lungs have completely collapsed (refer to figure 6 for the illustration of levels). The differential measurements at t₁ and t₂ again correspond to the measurements for the case of deflated and inflated lungs, respectively, however, this time with the collapse. AWGN of 60 dB was added to the voltage data. Figure 8 shows the reconstructed images for different cases. The left column shows the lung function which has happened in both lungs simultaneously. The middle and right columns represent the cases that the accumulation develops faster in one lung than the other, i.e., corresponding to the collapse in the left lung and right lung only. The rows denote different levels of collapse. Since the conductivity of the collapsed region is time-invariant, the amplitudes of the collapsed regions get smaller. It is possible to visually inspect the difference in the distributions due to the considerable drop in amplitude.

Fig. 8

It is also possible to image the edematous lung injury by recording the initial measurements from the patient with healthy lungs at t₁ and obtaining the differential measurements assuming that the fluid accumulation emerged at t₂. Again, AWGN corresponding to 40, 60 and 80 dB SNR levels was added to the voltage data. Figure 9 shows the reconstructed edematous regions for different levels of collapse. The intensity of the reconstructed perturbation gets larger when increasing the volume of the edematous region. The “spiky” artifacts are again visible near the surface.

Fig. 9

Figure 10, shows the normalized edema reconstructions along a line passing through the peak value of the reconstructed edema from ventral to dorsal, normal to the coil array plane, (see the red line in figure 9). The x-axis represents the ventral to dorsal orientation. The corresponding full-width-at-half-maximum (FWHM) values of the curves are illustrated as boxcar functions. It was found that the FWHMs provide marginal information on the level of the collapse, however, it is clearly visible that the amplitude drops drastically with the increasing volume of fluid accumulation. Thus, the variations in amplitude can be used to infer the development of fluid accumulation rather than the distributions.

Fig. 10

Fig. 11

It is inevitable that the patients change the posture or position during ventilation which causes alterations in the measurement signal. It was previously reported that these alterations are noticed as serious artifacts in the images [6]. To this end, we performed a simulation test to investigate the stability of image reconstructions to several body distorsions. The differential measurement were simulated by assuming that the body posture is modified at t₂ as opposed to the state at t₁. Figure 9 illustrates the possible artifacts that may be observed for different distorsions. The columns represent the images reconstructed from the voltage data with different SNR levels. The rows denote the images reconstructed from different body distorsions, i.e., 10% chest expansion, 10^o clockwise rotation and 2 cm right-lateral displacement. The expansion of chest does not cause too severe artifacts and the lungs can still be reconstructed seperately. However, the rotation and displacement provide very poor reconstructions with many artifacts.