**Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders**

Renato Toffanin1, Giuseppe Guglielmi2,3 and Maria A. Cova4 *1Advanced Research Centre for Health, Environment and Space (ARCHES) 2Dept. of Radiology, Casa Sollievo della Sofferenza, IRCCS 3Dept. of Radiology, University of Foggia 4Dept. of Radiology, University of Trieste Italy* 

## **1. Introduction**

238 Medical Imaging

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Evaluation of specific magnetic resonance (MR) parameters of the skeletal system holds great potential for the accurate clinical assessment of degenerative changes occurring in bone and soft tissues at different anatomical sites. MR imaging of the water in a joint can provide anatomical information about all the soft tissues within the synovial sac (articular cartilage, meniscus, ligaments, synovial fluid) and surrounding it (muscles, tendons, vascular structures). Importantly too, MRI of the water-plus-fat can provide information pertaining both to the bone density and trabecular architecture. Recent developments have led to combinations of scan protocols and image-measurement software such that MRI can be used to evaluate the spatial distribution of specific relaxation parameters and thus detect, assess, and quantify the many pathologic processes affecting the skeletal tissues. In the articular cartilage of knee, for example, gradual deterioration of the chondral tissue leads to progressive increases in the transverse relaxation time (T2) of the water protons (David-Vaudey, 2004; Dunn et al., 2004; Apprich et al., 2010). Similarly, bone loss in the calcaneus of patients with varying degrees of osteopenia and osteoporosis causes a prolongation of the effective transverse relaxation time (T2\*) of the bone marrow protons (Wehrli et al., 1995; Damilakis et al, 2004). Nonetheless, standard scan protocols for quantitative MRI are relatively slow and, therefore, not suitable for routine clinical applications. Faster methods would highly enhance their applicability in the clinical evaluation of skeletal disorders.

The purpose of this chapter is to provide a perspective on fast MRI methods for the noninvasive assessment of the skeletal status and their relevance to the diagnosis of osteoporosis and osteoarthritis, two major public health burdens (Hannan et al., 2001; Theis et al., 2007). The emphasis lies on echo-planar imaging (EPI)-based sequences (Tsao, 2010) for the accurate evaluation of pathologic processes affecting bone marrow and cartilage at different anatomical locations. A multi-shot EPI sequence is proposed for the fast T2\* mapping of the lumbar bone marrow while a gradient- and spin-echo (GRASE) sequence is suggested for the fast T2 mapping of patellar articular cartilage. The description of these fast acquisition techniques is followed by a presentation of two in

Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders 241

Fig. 2. A simplified diagram for the turbo spin-echo pulse sequence. RF: radiofrequency pulse, SS: slice selection gradient, PE: phase encoding gradient, FE: frequency encoding

This sequence is based on multi-echo multi-shot (MEMS) (Mehlkopf et al., 1984) and rapid acquisition with relaxation enhancement (RARE) (Hennig et al., 1986) sequences and provides T2-weighted images at fractions of the acquisition time of the conventional spinecho images. By applying multiple refocusing 180° RF pulses after the first echo, additional spin echoes can be generated. Between each successive echo, the phase-encoding gradients can be used to prepare the spins for different lines in k-space. Thus, multiple lines in k-space can be sampled per excitation. Each echo in the readout train is progressively weaker, as

Another method of decreasing image acquisition time is by echo-planar imaging (EPI) (Mansfield, 1977). In EPI multiple lines of k-space are acquired through a multiple-echo readout. However, in EPI signals are produced by rapid switching of gradient polarity in place of the slower selective 180° RF pulses. In this way, EPI can produce an image in less than 100 ms. However, in EPI sequences, since the multiple echoes are refocused by gradients and not by 180° pulses, there is more effect of T2\* decay and other artefacts. Therefore, both spin-echo and gradient-echo EPI sequences may be applied for the fast

Fig. 3. A simplified diagram for the GRASE (gradient- and spin-echo) pulse sequence

gradient

defined by the T2 decay.

evaluation of the T2\* relaxation.

vivo feasibility studies on a clinical 1.5 T MRI scanner. These investigations demonstrate that the proposed MRI methods can produce relaxation maps of specific skeletal sites in just a few minutes, and with mean values comparable with data obtained using conventional sequences. Their potential application in the clinical evaluation of osteoporosis and osteoarthritis is also discussed.

#### **2. Fast MRI techniques**

Transverse relaxation is the result of random interactions at the atomic and molecular levels (Abragam, 1961). This physical phenomenon is primarily related to the intrinsic field caused by adjacent protons (spins) and hence is called spin-spin relaxation. Transverse relaxation causes irreversible dephasing of the transverse magnetization. There is also a reversible bulk field dephasing effect caused by local field inhomogeneities, and its characteristic time is referred to as T2**\*** relaxation. These additional dephasing fields come from the main magnetic field inhomogeneity, the differences in magnetic susceptibility among various tissues or materials, chemical shift, and gradients applied for spatial encoding (Mugler, 2006). This dephasing can be eliminated by using a 180° pulse, as in a spin-echo sequence. Therefore, in a spin-echo sequence, only the 'true' T2 relaxation is seen. In gradient-echo sequences, there is no 180° refocusing pulse, and these dephasing effects are not eliminated. Hence, transverse relaxation in gradient-echo sequences (i.e., T2\* relaxation) is a combination of 'true' T2 relaxation and relaxation caused by magnetic field inhomogeneities.

Fig. 1. A diagram showing the T2 and T2\* relaxation decay curves

In order to obtain an accurate estimate of the transverse relaxation decay curves several images obtained at different echo times are generally required. The gold standard for T2 acquisition is likely to be a single slice single echo sequence (*i.e.* spin-echo sequence), repeated at several echo times, with long TR. A major improvement of the spin-echo technique is represented by the turbo spin-echo (TSE) sequence, of which a simplified diagram is depicted in Fig. 2.

vivo feasibility studies on a clinical 1.5 T MRI scanner. These investigations demonstrate that the proposed MRI methods can produce relaxation maps of specific skeletal sites in just a few minutes, and with mean values comparable with data obtained using conventional sequences. Their potential application in the clinical evaluation of

Transverse relaxation is the result of random interactions at the atomic and molecular levels (Abragam, 1961). This physical phenomenon is primarily related to the intrinsic field caused by adjacent protons (spins) and hence is called spin-spin relaxation. Transverse relaxation causes irreversible dephasing of the transverse magnetization. There is also a reversible bulk field dephasing effect caused by local field inhomogeneities, and its characteristic time is referred to as T2**\*** relaxation. These additional dephasing fields come from the main magnetic field inhomogeneity, the differences in magnetic susceptibility among various tissues or materials, chemical shift, and gradients applied for spatial encoding (Mugler, 2006). This dephasing can be eliminated by using a 180° pulse, as in a spin-echo sequence. Therefore, in a spin-echo sequence, only the 'true' T2 relaxation is seen. In gradient-echo sequences, there is no 180° refocusing pulse, and these dephasing effects are not eliminated. Hence, transverse relaxation in gradient-echo sequences (i.e., T2\* relaxation) is a combination of 'true' T2

relaxation and relaxation caused by magnetic field inhomogeneities.

Fig. 1. A diagram showing the T2 and T2\* relaxation decay curves

diagram is depicted in Fig. 2.

In order to obtain an accurate estimate of the transverse relaxation decay curves several images obtained at different echo times are generally required. The gold standard for T2 acquisition is likely to be a single slice single echo sequence (*i.e.* spin-echo sequence), repeated at several echo times, with long TR. A major improvement of the spin-echo technique is represented by the turbo spin-echo (TSE) sequence, of which a simplified

osteoporosis and osteoarthritis is also discussed.

**2. Fast MRI techniques** 

Fig. 2. A simplified diagram for the turbo spin-echo pulse sequence. RF: radiofrequency pulse, SS: slice selection gradient, PE: phase encoding gradient, FE: frequency encoding gradient

This sequence is based on multi-echo multi-shot (MEMS) (Mehlkopf et al., 1984) and rapid acquisition with relaxation enhancement (RARE) (Hennig et al., 1986) sequences and provides T2-weighted images at fractions of the acquisition time of the conventional spinecho images. By applying multiple refocusing 180° RF pulses after the first echo, additional spin echoes can be generated. Between each successive echo, the phase-encoding gradients can be used to prepare the spins for different lines in k-space. Thus, multiple lines in k-space can be sampled per excitation. Each echo in the readout train is progressively weaker, as defined by the T2 decay.

Another method of decreasing image acquisition time is by echo-planar imaging (EPI) (Mansfield, 1977). In EPI multiple lines of k-space are acquired through a multiple-echo readout. However, in EPI signals are produced by rapid switching of gradient polarity in place of the slower selective 180° RF pulses. In this way, EPI can produce an image in less than 100 ms. However, in EPI sequences, since the multiple echoes are refocused by gradients and not by 180° pulses, there is more effect of T2\* decay and other artefacts. Therefore, both spin-echo and gradient-echo EPI sequences may be applied for the fast evaluation of the T2\* relaxation.

Fig. 3. A simplified diagram for the GRASE (gradient- and spin-echo) pulse sequence

Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders 243

The use of T2\* relaxometry can certainly promote the application of quantitative MRI in the diagnosis of osteoporosis. Nonetheless, MRI protocols commonly applied to estimate T2\* in bone marrow are relatively slow and, therefore, not suitable for routine clinical application. In one recent study on the calcaneus of six healthy volunteers, Toffanin et al. (2006) demonstrated the possibility of ultrafast T2\* mapping of the bone marrow both at 1.5 and 3 T. To obtain an accurate estimate of T2\* at 3.0 T or higher magnetic fields, corrective measures may be required during postprocessing to minimise local field variations (ΔB0) responsible for signal loss and consequent overestimation of the R2\* relaxation rate (1/T2\*). In the method proposed by Dahnke and Schaeffter (2005), the main field heterogeneity is derived from T2\* calculated on more than one slice and is used as an initial value for interactive optimisation, with which the relaxation signal is corrected for

The feasibility of a multi-shot gradient-echo EPI sequence for the fast T2\* mapping of the lumbar bone marrow was evaluated by our research team on a commercial clinical 1.5 T MRI scanner located in the Department of Radiology of the Cattinara Hospital at the University of Trieste. The MRI trial was performed on 21 subjects (8 males and 13 females) referred to the hospital for low back pain. Five slices were acquired to image the lumbar spine in the sagittal plane and the L2 vertebral body in the axial plane. In both cases, a fast field-echo (FFE) multi-shot EPI sequence was applied with removed blip gradients in order to apply the same phase encoding to all gradient echoes The overall examination time was approximately 5 minutes. The main acquisition parameters are summarised in Table 1.

TR 400 ms 400 ms

TEmin 2.0 ms 2.0 ms

TEmax 15.8 ms 15.8 ms

EPI factor 25 25

Flip angle 30° 30°

FOV 300 mm 300 mm 200 mm 200 mm

Matrix 320 320 224 224

Slice thickness 5 mm 5 mm

Table 1. Acquisition parameters of the fast field-echo multi-shot EPI sequence used for fast

No. of slices 5 5

T2\* mapping of the lumbar bone marrow at 1.5 T

Sagittal plane Axial plane

each voxel.

**3.1 Fast T2\* mapping of the lumbar bone marrow** 

By combining the TSE and EPI methods, the GRASE (gradient- and spin-echo) sequence (Fig. 3) uses a train of refocusing 180° pulses, but for each spin-echo of the readout, there are additional gradient recall echoes (Feinberg & Oshio, 1991; Oshio & Feinberg, 1991). In this sequence, each successive spin-echo is progressively weaker, as defined by T2 decay whereas the strength of the gradient recalled echoes surrounding the spin-echo is defined by the T2\* decay envelope. By combining spin-echoes and short gradient-echo trains, the GRASE technique overcomes several potential problems of EPI, including large chemical shift, image distortions and signal loss from field inhomogeneity.

## **3. MRI of trabecular bone**

Even though bone cannot be evaluated with most of the available MRI techniques in that they are unable to generate sufficient signal, new quantitative MRI approaches are used to study trabecular bone density and structure (Wehrli et al., 2006; Majumdar, 2008). MRI can be used to evaluate trabecular bone in a number of skeletal sites, indirectly via the protons of the bone marrow. Indeed, the presence of the trabecular bone matrix affects the signal intensity of bone marrow, an effect that is particularly pronounced with certain MRI sequences. With respect to gradient-echo acquisitions, static magnetic field inhomogeneities produced by the difference between trabecular bone and neighbouring bone marrow cause a more rapid decay of the MRI signal, which can be quantified by measuring T2\*. Pioneering studies have shown that T2\* is correlated with trabecular bone density (Davis et al., 1986; Rosenthal et al., 1990), and therefore, the effective transverse relaxation (T2\*) is shorter in normal trabecular bone than in the less dense trabecular structures of osteoporotic bone tissue. It has also been shown that bone marrow T2\* reflects the orientation of the trabeculae and correlates with their mechanical strength (Chung et al., 1993; Jergas et al., 1995). These characteristics make MRI a fundamental tool in evaluating the quality of spongy bone and increase the ability of the technique not only in identifying occult fractures but also in making possible a more accurate prediction of fracture risk.

T2\* relaxometry has been conducted at several sites of both axial and peripheral skeleton (Funke et al., 1994; Grampp et al., 1995; Link et al., 1998). The preferred site for quantitative MRI studies is the calcaneus in that it is mostly composed of spongy bone (95%). Therefore quantitative MRI of the calcaneus is extremely sensitive in identifying changes in bone quality that are not revealed by bone mineral densitometry. In one MRI study at 1.5 T that examined 68 women with different degrees of vertebral deformity (Wehrli et al., 2002), it was demonstrated that of the various areas of the calcaneus examined, the subtalar region was best able to discriminate patients with fracture from those without. The authors of this study also demonstrated that the R2\* (1/T2\*) is sensitive to changes in bone quality that were not identified with BMD.

Trabecular bone is also prominent in the vertebral body (up to 90%). The spine certainly represents the most critical skeletal site for quantitative MRI since vertebral fractures are the most common type of osteoporotic fractures (Wasnisch, 1999). In one MRI investigation done at 1.5 T on a group of 54 postmenopausal women, T2\* mapping of the lumbar spine was shown to be capable of differentiating between healthy subjects and subjects with low energy fractures (Damilakis et al., 2004).

By combining the TSE and EPI methods, the GRASE (gradient- and spin-echo) sequence (Fig. 3) uses a train of refocusing 180° pulses, but for each spin-echo of the readout, there are additional gradient recall echoes (Feinberg & Oshio, 1991; Oshio & Feinberg, 1991). In this sequence, each successive spin-echo is progressively weaker, as defined by T2 decay whereas the strength of the gradient recalled echoes surrounding the spin-echo is defined by the T2\* decay envelope. By combining spin-echoes and short gradient-echo trains, the GRASE technique overcomes several potential problems of EPI, including large chemical

Even though bone cannot be evaluated with most of the available MRI techniques in that they are unable to generate sufficient signal, new quantitative MRI approaches are used to study trabecular bone density and structure (Wehrli et al., 2006; Majumdar, 2008). MRI can be used to evaluate trabecular bone in a number of skeletal sites, indirectly via the protons of the bone marrow. Indeed, the presence of the trabecular bone matrix affects the signal intensity of bone marrow, an effect that is particularly pronounced with certain MRI sequences. With respect to gradient-echo acquisitions, static magnetic field inhomogeneities produced by the difference between trabecular bone and neighbouring bone marrow cause a more rapid decay of the MRI signal, which can be quantified by measuring T2\*. Pioneering studies have shown that T2\* is correlated with trabecular bone density (Davis et al., 1986; Rosenthal et al., 1990), and therefore, the effective transverse relaxation (T2\*) is shorter in normal trabecular bone than in the less dense trabecular structures of osteoporotic bone tissue. It has also been shown that bone marrow T2\* reflects the orientation of the trabeculae and correlates with their mechanical strength (Chung et al., 1993; Jergas et al., 1995). These characteristics make MRI a fundamental tool in evaluating the quality of spongy bone and increase the ability of the technique not only in identifying occult fractures but also in making possible a more accurate prediction of

T2\* relaxometry has been conducted at several sites of both axial and peripheral skeleton (Funke et al., 1994; Grampp et al., 1995; Link et al., 1998). The preferred site for quantitative MRI studies is the calcaneus in that it is mostly composed of spongy bone (95%). Therefore quantitative MRI of the calcaneus is extremely sensitive in identifying changes in bone quality that are not revealed by bone mineral densitometry. In one MRI study at 1.5 T that examined 68 women with different degrees of vertebral deformity (Wehrli et al., 2002), it was demonstrated that of the various areas of the calcaneus examined, the subtalar region was best able to discriminate patients with fracture from those without. The authors of this study also demonstrated that the R2\* (1/T2\*) is sensitive to changes in bone quality that

Trabecular bone is also prominent in the vertebral body (up to 90%). The spine certainly represents the most critical skeletal site for quantitative MRI since vertebral fractures are the most common type of osteoporotic fractures (Wasnisch, 1999). In one MRI investigation done at 1.5 T on a group of 54 postmenopausal women, T2\* mapping of the lumbar spine was shown to be capable of differentiating between healthy subjects and subjects with low

shift, image distortions and signal loss from field inhomogeneity.

**3. MRI of trabecular bone**

fracture risk.

were not identified with BMD.

energy fractures (Damilakis et al., 2004).

The use of T2\* relaxometry can certainly promote the application of quantitative MRI in the diagnosis of osteoporosis. Nonetheless, MRI protocols commonly applied to estimate T2\* in bone marrow are relatively slow and, therefore, not suitable for routine clinical application. In one recent study on the calcaneus of six healthy volunteers, Toffanin et al. (2006) demonstrated the possibility of ultrafast T2\* mapping of the bone marrow both at 1.5 and 3 T. To obtain an accurate estimate of T2\* at 3.0 T or higher magnetic fields, corrective measures may be required during postprocessing to minimise local field variations (ΔB0) responsible for signal loss and consequent overestimation of the R2\* relaxation rate (1/T2\*). In the method proposed by Dahnke and Schaeffter (2005), the main field heterogeneity is derived from T2\* calculated on more than one slice and is used as an initial value for interactive optimisation, with which the relaxation signal is corrected for each voxel.

## **3.1 Fast T2\* mapping of the lumbar bone marrow**

The feasibility of a multi-shot gradient-echo EPI sequence for the fast T2\* mapping of the lumbar bone marrow was evaluated by our research team on a commercial clinical 1.5 T MRI scanner located in the Department of Radiology of the Cattinara Hospital at the University of Trieste. The MRI trial was performed on 21 subjects (8 males and 13 females) referred to the hospital for low back pain. Five slices were acquired to image the lumbar spine in the sagittal plane and the L2 vertebral body in the axial plane. In both cases, a fast field-echo (FFE) multi-shot EPI sequence was applied with removed blip gradients in order to apply the same phase encoding to all gradient echoes The overall examination time was approximately 5 minutes. The main acquisition parameters are summarised in Table 1.


Table 1. Acquisition parameters of the fast field-echo multi-shot EPI sequence used for fast T2\* mapping of the lumbar bone marrow at 1.5 T

Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders 245

Fig. 5. Overview of the T2\* data previously obtained with conventional sequences

proposed approach in the clinical evaluation of osteoporosis.

introduction of T2 mapping in routine clinical protocols.

**4. MRI of articular cartilage**

2000; Mosher & Dardzinski, 2004).

These results indicate that fast T2\* mapping of the lumbar bone marrow is feasible on a 1.5 T scanner. However, further studies are required to investigate the full potential of the

Articular cartilage, is one of the types of hyaline cartilage that persists throughout adult life. Basically, it comprises chondrocytes incorporated in an extracellular matrix composed mainly of water, collagen II fibrils and proteoglycans (Seibel et al., 2004). Despite its simple appearance, this tissue hides various modifications in respect of the original cartilage that make it a singular structure. The articular cartilage is, in fact, stratified and classically, four distinct layers are described from the surface to the interior: tangential, transitional, radial and calcified, respectively (Fig. 6). Both morphological and biochemical information can be obtained by MRI, which is probably the most accurate imaging modality in evaluating the state of hyaline cartilage (Disler et al., 2000; Cova & Toffanin, 2002). Apart from clinical MRI protocols that depict cartilage morphology, there is a growing interest in developing quantitative MRI approaches that are sensitive to its early structural changes (Burstein et al.,

Over the past years, quantification of the human articular cartilage has been performed using T1, T1ρ and T2 relaxation time constants as well as the magnetization transfer ratio (Toffanin et al., 2001; Menezes et al., 2004; Wheaton et al., 2005). One of the magnetic parameter that is currently evaluated for studying cartilage damage is the transverse relaxation time (T2), whose relaxation mechanism results dominated by the dipolar interaction between water molecules and collagen (Mlynárik et al., 2004) In this regard, dual-echo or multi-echo sequences are typically employed for quantitative T2 mapping. Nonetheless, faster quantitative MRI techniques are required in order to allow the

Estimation of the T2\* relaxation time was performed on one manually-defined region drawn on the entire L2 vertebral body as shown in Fig. 4. The multi-shot EPI sequence produced T2\* maps with mean values comparable with previous data obtained with conventional sequences (Fig. 5). The mean T2\* measured in the sagittal plane (14.2 ± 3.9 ms) was slightly lower than that measured in the axial plane (14.7 ±3.9 ms) but no statistically significant difference was observed (*P* < 0.05).

Fig. 4. T2\* maps of the central slice of the L2 vertebra generated from sagittal (a) and axial (b) images by means of a monoexponential fitting algorithm as described by Dahnke & Schaeffter (2005). The mean T2\* was measured over the entire vertebral body excluding the cortical bone

Estimation of the T2\* relaxation time was performed on one manually-defined region drawn on the entire L2 vertebral body as shown in Fig. 4. The multi-shot EPI sequence produced T2\* maps with mean values comparable with previous data obtained with conventional sequences (Fig. 5). The mean T2\* measured in the sagittal plane (14.2 ± 3.9 ms) was slightly lower than that measured in the axial plane (14.7 ±3.9 ms) but no statistically significant

(a)

(b) Fig. 4. T2\* maps of the central slice of the L2 vertebra generated from sagittal (a) and axial (b) images by means of a monoexponential fitting algorithm as described by Dahnke & Schaeffter (2005). The mean T2\* was measured over the entire vertebral body excluding the

difference was observed (*P* < 0.05).

cortical bone


Fig. 5. Overview of the T2\* data previously obtained with conventional sequences

These results indicate that fast T2\* mapping of the lumbar bone marrow is feasible on a 1.5 T scanner. However, further studies are required to investigate the full potential of the proposed approach in the clinical evaluation of osteoporosis.

## **4. MRI of articular cartilage**

Articular cartilage, is one of the types of hyaline cartilage that persists throughout adult life. Basically, it comprises chondrocytes incorporated in an extracellular matrix composed mainly of water, collagen II fibrils and proteoglycans (Seibel et al., 2004). Despite its simple appearance, this tissue hides various modifications in respect of the original cartilage that make it a singular structure. The articular cartilage is, in fact, stratified and classically, four distinct layers are described from the surface to the interior: tangential, transitional, radial and calcified, respectively (Fig. 6). Both morphological and biochemical information can be obtained by MRI, which is probably the most accurate imaging modality in evaluating the state of hyaline cartilage (Disler et al., 2000; Cova & Toffanin, 2002). Apart from clinical MRI protocols that depict cartilage morphology, there is a growing interest in developing quantitative MRI approaches that are sensitive to its early structural changes (Burstein et al., 2000; Mosher & Dardzinski, 2004).

Over the past years, quantification of the human articular cartilage has been performed using T1, T1ρ and T2 relaxation time constants as well as the magnetization transfer ratio (Toffanin et al., 2001; Menezes et al., 2004; Wheaton et al., 2005). One of the magnetic parameter that is currently evaluated for studying cartilage damage is the transverse relaxation time (T2), whose relaxation mechanism results dominated by the dipolar interaction between water molecules and collagen (Mlynárik et al., 2004) In this regard, dual-echo or multi-echo sequences are typically employed for quantitative T2 mapping. Nonetheless, faster quantitative MRI techniques are required in order to allow the introduction of T2 mapping in routine clinical protocols.

Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders 247

Fig. 7. a–h. Axial GRASE images of patella cartilage (TR/TEmin-TEmax: 3,000/15–120 ms); a: 15 ms, b: 30 ms, c: 45 ms, d: 60 ms, e: 75 ms, f: 90 ms, g: 105 ms, h: 120 ms. There is a clear decay in the signal intensity of the patellar articular cartilage at longer TE, which can be visually observed. The T2 relaxation time constant was calculated from a linear least-square fit to the logarithm of the image intensity data (From Quaia et al., 2008. Reprinted with

permission).

Fig. 6. Histological zones in hyaline cartilage. Collagen fibres are parallel to the surface in the superficial (tangential) zone, curved in the intermediate (transitional) zone and perpendicular to subchondral bone in the deep (radial) zone (From Cova & Toffanin, 2002. Reprinted with permission).

## **4.1 Fast T2 mapping of the patellar articular cartilage**

Recently, we have devoted particular attention to optimising specific quantitative MRI protocols for the fast T2 mapping of knee cartilage. The focus was on the gradient- and spinecho (GRASE) sequence able to produce a set of T2-weighted images in less than 2 minutes. Also this research study was conducted on a commercial clinical 1.5 T MRI scanner located in the Department of Radiology of the Cattinara Hospital at the University of Trieste. The feasibility of the proposed approach was assessed on 35 patients (21 males and 14 females) with moderate degree of patellar osteoarthritis. (Quaia et al., 2008).

For each patient, transverse GRASE and TSE images of patellar cartilage were acquired using the scan protocols summarised in Table 2.


Table 2. Acquisition parameters of the GRASE and TSE sequences used for fast T2 mapping of the patellar articular cartilage at 1.5 T

Fig. 6. Histological zones in hyaline cartilage. Collagen fibres are parallel to the surface in the superficial (tangential) zone, curved in the intermediate (transitional) zone and

perpendicular to subchondral bone in the deep (radial) zone (From Cova & Toffanin, 2002.

Recently, we have devoted particular attention to optimising specific quantitative MRI protocols for the fast T2 mapping of knee cartilage. The focus was on the gradient- and spinecho (GRASE) sequence able to produce a set of T2-weighted images in less than 2 minutes. Also this research study was conducted on a commercial clinical 1.5 T MRI scanner located in the Department of Radiology of the Cattinara Hospital at the University of Trieste. The feasibility of the proposed approach was assessed on 35 patients (21 males and 14 females)

For each patient, transverse GRASE and TSE images of patellar cartilage were acquired

TR 3,000 ms 3,000 ms TEmin 15 ms 15 ms TEmax 120 ms 120 ms EPI factor 3 - Turbo factor 8 8

FOV 80 mm × 80 mm 80 mm × 80 mm Matrix 128 × 128 128 × 128 Slice thickness 3 mm 3 mm No. of slices 10 10 Total scan time 1 min 51 s 5 min 52 s Table 2. Acquisition parameters of the GRASE and TSE sequences used for fast T2 mapping

**GRASE TSE** 

Reprinted with permission).

**4.1 Fast T2 mapping of the patellar articular cartilage** 

using the scan protocols summarised in Table 2.

of the patellar articular cartilage at 1.5 T

with moderate degree of patellar osteoarthritis. (Quaia et al., 2008).

Fig. 7. a–h. Axial GRASE images of patella cartilage (TR/TEmin-TEmax: 3,000/15–120 ms); a: 15 ms, b: 30 ms, c: 45 ms, d: 60 ms, e: 75 ms, f: 90 ms, g: 105 ms, h: 120 ms. There is a clear decay in the signal intensity of the patellar articular cartilage at longer TE, which can be visually observed. The T2 relaxation time constant was calculated from a linear least-square fit to the logarithm of the image intensity data (From Quaia et al., 2008. Reprinted with permission).

Fast MRI Methods for the Clinical Evaluation of Skeletal Disorders 249

In our series the GRASE sequence provided T2 values slightly lower than those obtained by the TSE sequence in most patients. This may relate to T2\* decay present in the GRASE sequence. Indeed, a potential drawback of the GRASE sequence is the possible presence of artefacts due to the echo-planar imaging readout module that may lead to pronounced chemical shift, image distortion, and signal loss from magnetic field inhomogeneity. Nonetheless, differences in T2 values within deep and superficial cartilage were in line with those observed with other MR sequences (Van Breuseghem et al., 2004). They are related to cartilage T2 anisotropy determined by the different direction of the collagen fibres in the superficial and deep cartilage with respect to the static magnetic field (Mosher et al., 2001).

The above MRI methods can drastically reduce the scan time for measuring the spatial distribution of specific relaxation parameters in bone and cartilage. They may become more widely adopted, being applied either with other imaging techniques or in isolation, to better evaluate skeletal disorders, identify early tissue degeneration, tailor therapeutic

This work was supported in part by a grant from the Italian Ministry for Education,

Abragam, A. (1983). *The Principles of Nuclear Magnetism* (2nd edition), Oxford University

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Cova, M. & Toffanin, R. (2002). MR microscopy of hyaline cartilage: current status. *European* 

Dahnke, H. & Schaeffter, T. (2005). Limits of detection of SPIO at 3.0 T using T2\*

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*Osteoarthritis and Cartilage*, Vol.18, No.9, pp. 1211-1217, ISSN 1063-4584 Burstein, D.; Bashir, A. & Gray, M.L. (2000). MRI techniques in early stages of cartilage disease. *Investigative Radiology*, Vol.35, No.10, pp. 622-638, ISSN 0020-9996 Chung, H.; Wehrli, F.W.; Williams, J.L. & Kugelmass, S.D. (1993). Relationship between

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relaxometry. *Magnetic Resonance in Medicine*, Vol.53, No.5, pp 1202-1206, ISSN

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**5. Conclusion**

**6. Acknowledgment**

0027-8424

0740-3194

**7. References**

interventions and follow treatment response.

University and Research (5 per mille 2009).

Press, ISBN 978-0-19-852014-6, New York

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No.11, pp. 706-712, ISSN 0020-9996

Estimation of T2 was performed on one manually-defined region drawn on the entire patella cartilage as shown in Fig. 8.

Fig. 8. Axial GRASE image of patella cartilage (TR/TE: 3,000/15 ms). A manually defined ROI is drawn on the entire patellar articular cartilage for the quantification of the global T2 relaxation time. The ROI includes the entire cartilage, encompassing both the deep cartilage close to the subchondral bone and the superficial cartilage, and the edges of the joint surface (From Quaia et al., 2008. Reprinted with permission).


Table 3. Mean T2 values for patellar articular cartilage obtained from the GRASE and TSE images of the patellar articular cartilage of selected patients together with the corresponding arthroscopic grading1.

<sup>1</sup> Osteoarthritis from anteroposterior radiographs was graded according to the Kellgren–Lawrence scoring system (0=no osteoarthritic features; 1=minute osteophytes of doubtful importance; 2=definite osteophytes without reduction of the joint space; 3=reduction of the joint space; 4=greatly reduced joint space and sclerosis of the subchondral bone) (Kellgren & Lawrence, 1957).

In our series the GRASE sequence provided T2 values slightly lower than those obtained by the TSE sequence in most patients. This may relate to T2\* decay present in the GRASE sequence. Indeed, a potential drawback of the GRASE sequence is the possible presence of artefacts due to the echo-planar imaging readout module that may lead to pronounced chemical shift, image distortion, and signal loss from magnetic field inhomogeneity. Nonetheless, differences in T2 values within deep and superficial cartilage were in line with those observed with other MR sequences (Van Breuseghem et al., 2004). They are related to cartilage T2 anisotropy determined by the different direction of the collagen fibres in the superficial and deep cartilage with respect to the static magnetic field (Mosher et al., 2001).

## **5. Conclusion**

248 Medical Imaging

Estimation of T2 was performed on one manually-defined region drawn on the entire

Fig. 8. Axial GRASE image of patella cartilage (TR/TE: 3,000/15 ms). A manually defined ROI is drawn on the entire patellar articular cartilage for the quantification of the global T2 relaxation time. The ROI includes the entire cartilage, encompassing both the deep cartilage close to the subchondral bone and the superficial cartilage, and the edges of the joint surface

Patient No. GRASE TSE Arthroscopic grading 1 44.0±10.0 43.5±8.0 2B 2 48.3±8.0 47.2±6.0 2B 3 25.5±9.0 26.3±7.5 0 4 38.1±7.0 38.4±6.0 1 7 45.2±6.0 46.3±7.0 3A 12 30.5±8.0 31.0±6.0 2A 18 35.5±8.0 37.2±8.0 1 20 39.2±7.0 38.5±6.0 2A 27 28.5±6.0 29.0±9.0 0 28 58.2±6.0 57.2±11.0 3A 35 61.2±11.0 64.3±10.0 3B Table 3. Mean T2 values for patellar articular cartilage obtained from the GRASE and TSE images of the patellar articular cartilage of selected patients together with the corresponding

1 Osteoarthritis from anteroposterior radiographs was graded according to the Kellgren–Lawrence scoring system (0=no osteoarthritic features; 1=minute osteophytes of doubtful importance; 2=definite osteophytes without reduction of the joint space; 3=reduction of the joint space; 4=greatly reduced joint

space and sclerosis of the subchondral bone) (Kellgren & Lawrence, 1957).

patella cartilage as shown in Fig. 8.

(From Quaia et al., 2008. Reprinted with permission).

arthroscopic grading1.

The above MRI methods can drastically reduce the scan time for measuring the spatial distribution of specific relaxation parameters in bone and cartilage. They may become more widely adopted, being applied either with other imaging techniques or in isolation, to better evaluate skeletal disorders, identify early tissue degeneration, tailor therapeutic interventions and follow treatment response.

## **6. Acknowledgment**

This work was supported in part by a grant from the Italian Ministry for Education, University and Research (5 per mille 2009).

## **7. References**


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**0**

**12**

*Brazil*

**Determination of Cardiac Ejection Fraction by**

Franciane C. Peters1, Luis Paulo da S. Barra2 and Rodrigo W. dos Santos<sup>2</sup>

Cardiac ejection fraction is a clinical parameter that infers the efficiency of the heart as a pump. The ejection fraction of left ventricle (EFLV) and right ventricle (EFRV) are determined separately. However, the clinical use of EFLV is more common and it is frequently called ejection fraction (EF). By definition, the ejection fraction is calculated in the following way:

*EDV* <sup>=</sup> *EDV* <sup>−</sup> *ESV*

where *PV* is the volume of blood pumped, that is given by the difference between the end-diastolic volume (*EDV*) and the end-systolic volume (*ESV*). Cardiac ejection fraction is a relevant parameter because it is highly correlated to the functional status of the heart. To determine the EF, different non-invasive techniques can be used, like echocardiography, cardiac magnetic resonance and computed tomography. However, they are not suitable for continuous monitoring. In this work, we numerically evaluate a new method for the continuous estimation of cardiac ejection fraction based on Electrical Impedance Tomography

EIT is a technique that reconstructs an image of the electrical resistivity inside a domain based on protocols of current injection and potential measurement on the external boundary of the domain. Detailed description about the development of this technique can be found, for instance, in Cheney et al. (1999) and Holder (2005). The EIT has a large utilization on geophysics and monitoring of industrial activities (Kim et al., 2004; Park et al., 2008), as non-destructive technique to evaluate structures (Karhunen et al., 2009; Peters et al., 2010) and on biomedical engineering (Barber & Brown, 1984; Brown et al., 1985; Mello et al., 2008; Trigo et al., 2004). In this last area, the EIT has been developed to detect breast and other kinds of cancer (Choi et al., 2007; Seo et al., 2004) and to monitor lung ventilation (Adler et al., 1997; Lima et al., 2007), brain activity (Polydorides et al., 2002), heart activity (Peters et al.,

The special interest of the biomedical engineering in the development of the EIT is due to its safety for monitoring long periods, since it is not based on ionizing radiation. On the other hand, the EIT spacial resolution is not as high as the traditional imaging methods.

*EF* <sup>=</sup> *PV*

**1. Introduction**

(EIT).

2009; Zlochiver et al., 2006), among others.

**Electrical Impedance Tomography**

<sup>1</sup>*Federal University of Rio de Janeiro* <sup>2</sup>*Federal University of Juiz de Fora*

*EDV* , (1)

(4th edition), Murray, F.J., Editor, pp. 257–259, Lippincott, Williams & Wilkins, ISBN 978-0781720380, Philadelphia


## **Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography**

Franciane C. Peters1, Luis Paulo da S. Barra2 and Rodrigo W. dos Santos<sup>2</sup> <sup>1</sup>*Federal University of Rio de Janeiro* <sup>2</sup>*Federal University of Juiz de Fora Brazil*

#### **1. Introduction**

252 Medical Imaging

Wehrli, F.W.; Hilaire, L.; Fernández-Seara, M; Gomberg, B.R.; Song, H.K.; Zemel, B.; Loh, L. &

*Journal of Bone Mineral Research*, Vol.17, No.12, pp. 2265–2273, ISSN 0884-0431 Wehrli, F.W.; Ford, J.C. & Haddad, J.G. (1995). Osteoporosis: clinical assessment wit

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Wheaton, A.J.; Dodge, G.R.; Borthakur, A.; Kneeland, J.B.; Schumacher, H.R. & Reddy, R.

ISBN 978-0781720380, Philadelphia

0033-8419

0736-0266

731–776, ISSN 0952-3480

(4th edition), Murray, F.J., Editor, pp. 257–259, Lippincott, Williams & Wilkins,

Snyder, P.J. (2002). Quantitative magnetic resonance imaging in the calcaneus and femur of women with varying degrees of osteopenia and vertebral deformity status.

quantitative MR imaging in diagnosis. *Radiology*, Vol.196, No.3, pp. 631-641, ISSN

assessment of bone structure and function. *NMR in Biomedicine*, Vol.19, No.7, pp.

(2005). Detection of changes in articular cartilage proteoglycan by T1(rho) magnetic resonance imaging. *Journal of Orthopaedic Research*, Vol.23, No.1, pp. 102–108, ISSN

> Cardiac ejection fraction is a clinical parameter that infers the efficiency of the heart as a pump. The ejection fraction of left ventricle (EFLV) and right ventricle (EFRV) are determined separately. However, the clinical use of EFLV is more common and it is frequently called ejection fraction (EF). By definition, the ejection fraction is calculated in the following way:

$$EF = \frac{PV}{EDV} = \frac{EDV - ESV}{EDV} \,\,\,\,\tag{1}$$

where *PV* is the volume of blood pumped, that is given by the difference between the end-diastolic volume (*EDV*) and the end-systolic volume (*ESV*). Cardiac ejection fraction is a relevant parameter because it is highly correlated to the functional status of the heart. To determine the EF, different non-invasive techniques can be used, like echocardiography, cardiac magnetic resonance and computed tomography. However, they are not suitable for continuous monitoring. In this work, we numerically evaluate a new method for the continuous estimation of cardiac ejection fraction based on Electrical Impedance Tomography (EIT).

EIT is a technique that reconstructs an image of the electrical resistivity inside a domain based on protocols of current injection and potential measurement on the external boundary of the domain. Detailed description about the development of this technique can be found, for instance, in Cheney et al. (1999) and Holder (2005). The EIT has a large utilization on geophysics and monitoring of industrial activities (Kim et al., 2004; Park et al., 2008), as non-destructive technique to evaluate structures (Karhunen et al., 2009; Peters et al., 2010) and on biomedical engineering (Barber & Brown, 1984; Brown et al., 1985; Mello et al., 2008; Trigo et al., 2004). In this last area, the EIT has been developed to detect breast and other kinds of cancer (Choi et al., 2007; Seo et al., 2004) and to monitor lung ventilation (Adler et al., 1997; Lima et al., 2007), brain activity (Polydorides et al., 2002), heart activity (Peters et al., 2009; Zlochiver et al., 2006), among others.

The special interest of the biomedical engineering in the development of the EIT is due to its safety for monitoring long periods, since it is not based on ionizing radiation. On the other hand, the EIT spacial resolution is not as high as the traditional imaging methods.

In this section, all the methods used to solve the EIT problem will be described. First, the parameterization based on magnetic resonance images will be explained, as well as the resistivity model of the thorax. Second, the governing equations of the forward problem will be introduced in addition to the numerical methods used to solve it. Third, the inverse problem will be formulated as a minimization problem and the adopted optimization method

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 255

From magnetic resonance (MR) images, the regions of interest, in this case the two ventricles, were manually segmented in two different phases: end of the systole and the end of diastole. Each curve of the segmentation was parameterized by a spline, with a minimum number of points. The left ventricle spline has 7 control points and the right one 8 points. The external boundary of the thorax and the boundaries of the lungs were also segmented. For simplicity, the shape and size of the thorax and the lungs are assumed constant during the heart cycle.

Fig. 1. Manual segmentation of an MR image. The boundaries of the lungs, ventricle cavities and thorax are represented by splines. LV and RV denotes left ventricle and right ventricle,

The goal of our method is to recover the shape of the internal cavities of the heart, presently considered in two dimensions, from electric potential measures. Therefore, with two coordinates for each control point of the spline, the methods would need to estimate a total of 30 ((7 + 8) × 2) parameters. To reduce the number of parameters in half, another parameterization is used. In this, only one parameter defines the position of each control

During MRI segmentation we have used the same number of control points for the splines in both systolic and diastolic phase. This allows us to restrict the search space forcing that each control point *i* belongs to a line that connects its position at systole and diastole, as shown in

A linear interpolation, parameterized by a scalar *ti*, is used between the values of the coordinates of each control point *i*. The spline relative to the end of systole can be obtained

will be presented. Finally, the computational experiments will be reported.

**2.1 Two-dimensional models based on magnetic resonance images**

**2.1.1 Parameterization**

respectively.

point.

Fig. 2.

Figure 1 shows a segmentation example.

Nevertheless, its portability, low cost and time resolution are the main advantages of the technique.

Mathematically, the EIT is classified as a non-linear inverse problem. Inverse because one wants to find the electrical resistivity of the body using measures of electrical potential on the boundary excited by known electrical current. The forward (or direct) problem related to the inverse one is to compute the electrical potential with known body resistivity and injected current. The inverse problem is non-linear, what means that there is not a linear relation between electrical resistivity and the electrical potential on the boundary. So, in general, the solution process starts with an estimated resistivity distribution and such estimative is iteratively adjusted in order to achieve an acceptable approximation for the actual resistivity distribution.

Furthermore, the inverse problem is ill-posed and ill-conditioned. In general, the number of parameters needed to represent the resistivity distribution is greater than the number of potential measures. So, in order to treat the ill-posedness of the problem, some strategies of regularization should be implemented. For instance, via the inclusion of previously known information about the resistivity distribution in the solution of the inverse problem. The problem is considered ill-conditioned because small perturbations in the measures can cause a large change in the found resistivity distribution. So, the process of image generation by EIT is very sensible to noise in the potential measures.

It is possible to see that many aspects are involved in the solution of the EIT problem and some of them were discussed in previous works of the same authors. Barra et al. (2006a) and Barra et al. (2006b) treat some computational aspects of the solution process. Peters & Barra (2009) treats the special kind of regularization also adopted here. Peters & Barra (2010) compares different measurement protocols and addresses the sensitivity of the process in the presence of noisy measures. Peters et al. (2009) presents the viability analysis of a biomedical application of the EIT. So the aim of this work is to revisit this particular biomedical application, describing all the procedures involved in the generation of a computational model based on cardiac magnetic ressonance images that allows the determination of the cardiac ejection fraction by the EIT. Preliminary results are presented and the potentialities and limitations of the proposed technique are discussed. The results suggest the proposed technique is a promising diagnostic tool that offers continuous and non-invasive estimation of cardiac ejection fraction.

#### **2. Methods**

Usual strategies to generate the EIT image is based on the discretization of the body in small parts (Borsic et al., 2001). Each part has an unknown parameter, its resistivity. So, to get a good image resolution, a great number of parameters is needed and the problem becomes ill-posed. In this strategy, regularization techniques are applied and after solving the inverse problem, the image obtained is modified by a limiarization process in order to get the final image.

In this work we adopted a different strategy to generate the EIT images. In order to monitor the cardiac ejection fraction, we assume that recent magnetic resonance images of the patient are available. So, this previous information allows the use of a different kind of parameterization in which the number of parameters is greatly reduced.

In this section, all the methods used to solve the EIT problem will be described. First, the parameterization based on magnetic resonance images will be explained, as well as the resistivity model of the thorax. Second, the governing equations of the forward problem will be introduced in addition to the numerical methods used to solve it. Third, the inverse problem will be formulated as a minimization problem and the adopted optimization method will be presented. Finally, the computational experiments will be reported.

## **2.1 Two-dimensional models based on magnetic resonance images**

## **2.1.1 Parameterization**

2 Will-be-set-by-IN-TECH

Nevertheless, its portability, low cost and time resolution are the main advantages of the

Mathematically, the EIT is classified as a non-linear inverse problem. Inverse because one wants to find the electrical resistivity of the body using measures of electrical potential on the boundary excited by known electrical current. The forward (or direct) problem related to the inverse one is to compute the electrical potential with known body resistivity and injected current. The inverse problem is non-linear, what means that there is not a linear relation between electrical resistivity and the electrical potential on the boundary. So, in general, the solution process starts with an estimated resistivity distribution and such estimative is iteratively adjusted in order to achieve an acceptable approximation for the actual resistivity

Furthermore, the inverse problem is ill-posed and ill-conditioned. In general, the number of parameters needed to represent the resistivity distribution is greater than the number of potential measures. So, in order to treat the ill-posedness of the problem, some strategies of regularization should be implemented. For instance, via the inclusion of previously known information about the resistivity distribution in the solution of the inverse problem. The problem is considered ill-conditioned because small perturbations in the measures can cause a large change in the found resistivity distribution. So, the process of image generation by EIT

It is possible to see that many aspects are involved in the solution of the EIT problem and some of them were discussed in previous works of the same authors. Barra et al. (2006a) and Barra et al. (2006b) treat some computational aspects of the solution process. Peters & Barra (2009) treats the special kind of regularization also adopted here. Peters & Barra (2010) compares different measurement protocols and addresses the sensitivity of the process in the presence of noisy measures. Peters et al. (2009) presents the viability analysis of a biomedical application of the EIT. So the aim of this work is to revisit this particular biomedical application, describing all the procedures involved in the generation of a computational model based on cardiac magnetic ressonance images that allows the determination of the cardiac ejection fraction by the EIT. Preliminary results are presented and the potentialities and limitations of the proposed technique are discussed. The results suggest the proposed technique is a promising diagnostic tool that offers continuous and non-invasive estimation

Usual strategies to generate the EIT image is based on the discretization of the body in small parts (Borsic et al., 2001). Each part has an unknown parameter, its resistivity. So, to get a good image resolution, a great number of parameters is needed and the problem becomes ill-posed. In this strategy, regularization techniques are applied and after solving the inverse problem, the image obtained is modified by a limiarization process in order to get the final image.

In this work we adopted a different strategy to generate the EIT images. In order to monitor the cardiac ejection fraction, we assume that recent magnetic resonance images of the patient are available. So, this previous information allows the use of a different kind of

parameterization in which the number of parameters is greatly reduced.

technique.

distribution.

is very sensible to noise in the potential measures.

of cardiac ejection fraction.

**2. Methods**

From magnetic resonance (MR) images, the regions of interest, in this case the two ventricles, were manually segmented in two different phases: end of the systole and the end of diastole. Each curve of the segmentation was parameterized by a spline, with a minimum number of points. The left ventricle spline has 7 control points and the right one 8 points. The external boundary of the thorax and the boundaries of the lungs were also segmented. For simplicity, the shape and size of the thorax and the lungs are assumed constant during the heart cycle. Figure 1 shows a segmentation example.

Fig. 1. Manual segmentation of an MR image. The boundaries of the lungs, ventricle cavities and thorax are represented by splines. LV and RV denotes left ventricle and right ventricle, respectively.

The goal of our method is to recover the shape of the internal cavities of the heart, presently considered in two dimensions, from electric potential measures. Therefore, with two coordinates for each control point of the spline, the methods would need to estimate a total of 30 ((7 + 8) × 2) parameters. To reduce the number of parameters in half, another parameterization is used. In this, only one parameter defines the position of each control point.

During MRI segmentation we have used the same number of control points for the splines in both systolic and diastolic phase. This allows us to restrict the search space forcing that each control point *i* belongs to a line that connects its position at systole and diastole, as shown in Fig. 2.

A linear interpolation, parameterized by a scalar *ti*, is used between the values of the coordinates of each control point *i*. The spline relative to the end of systole can be obtained

with

and

in Fig. 3.

P0

(1995).

other issues.

**2.1.3 Resistivity model**

*<sup>u</sup>*<sup>0</sup> <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*<sup>+</sup>

P1 P2

P3

(a) Shape of the curve

*k tk* <sup>−</sup> *<sup>T</sup>*<sup>+</sup> *k*

, *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*<sup>+</sup>

 *tk* <sup>−</sup> *<sup>T</sup>*<sup>+</sup> *k*

*tk*<sup>+</sup><sup>2</sup> − *T*<sup>−</sup>

P4 P5

*pk*−<sup>1</sup> = <sup>2</sup>

each control point **<sup>P</sup>***<sup>k</sup>* this new parameter affects the values *<sup>T</sup>*<sup>+</sup>

*T*<sup>+</sup>

*pk*<sup>+</sup><sup>1</sup> = 2

*k*+1 *tk*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>T</sup>*<sup>+</sup>

*k*+1

*k*+2 2

*<sup>k</sup>*−<sup>1</sup> <sup>=</sup> *tk* <sup>+</sup> *sk* , *<sup>T</sup>*<sup>−</sup>

<sup>2</sup> , *pk* <sup>=</sup> <sup>2</sup>

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 257

The flexibility of the model is improved by the introduction of new degree of freedom *sk*. In

In other words, one can say that the parameter *s* controls the distance between the curve and the control point, what allows the approximation or the interpolation of these points, as shown

P6

Fig. 3. The values of the parameters of each control point are: *s*<sup>0</sup> = 0, *s*<sup>1</sup> = 1, *s*<sup>2</sup> = 0, *s*<sup>3</sup> = 0, *s*<sup>4</sup> = 0, *s*<sup>5</sup> = 1, *s*<sup>6</sup> = 0. These figures were inspired by the work of Blanc & Schlick (1995).

Although the expressions used to implement the Extended X-spline were rewritten above, detailed description about this and other types of splines can be found in Blanc & Schlick

The proposed 2D model splits the domain in regions that represent different biological tissues, heart cavities, lungs and torso, each one associated with a different electrical resistivity.

Grimnes (2008) presents the main factors that influence the properties of biological tissues. Although they may be classified in only four groups ( epithelium, muscle, connective tissue and nervous tissue), the tissues can be divided in thirty kinds in accordance to their electrical properties (Gabriel, 1996). In addition, the value of the resistivity of each tissue depends on the frequency of the electrical current, on the temperature, on the presence of water, among

In this work, we assume the resistivity of each tissue as known, constant and isotropic. These are all simplified assumptions, since biological tissues are usually heterogeneous and

F(t)

0

0.5

1

F0

, *<sup>u</sup>*<sup>2</sup> <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*<sup>−</sup>

, *pk*<sup>+</sup><sup>2</sup> = 2

*k*+2 *tk*<sup>+</sup><sup>2</sup> − *T*<sup>−</sup>

*tk*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>T</sup>*<sup>+</sup>

*k*+2

*k*+1 2

*<sup>k</sup>*−<sup>1</sup> <sup>e</sup> *<sup>T</sup>*<sup>−</sup>

F1 F2

*k*+3 2

*tk*<sup>+</sup><sup>3</sup> − *T*<sup>−</sup>

, *<sup>u</sup>*<sup>3</sup> <sup>=</sup> *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*<sup>−</sup>

*<sup>k</sup>*+<sup>1</sup> as follows:

0 1 2 3 4 5 6

F3

t

(b) Blending functions

*<sup>k</sup>*+<sup>1</sup> = *tk* − *sk* (9)

*k*+3 *tk*<sup>+</sup><sup>3</sup> − *T*<sup>−</sup>

*k*+3

, (7)

. (8)

F4 F5

F6

(6)

Fig. 2. The control points are represented by red crosses. The dashed lines are the search space.

with *ti* = 0 for all *i*, and the one relative to the end of diastole with *ti* = 1 for all *i*. Doing so, the method goal is to recover the shape of the ventricular cavities via the estimation of the 15 parameters *ti*, with *i* = 1, 2, ..., 15.

#### **2.1.2 Splines**

Splines are mathematical models that associate a curve with a set of points named control points. Here we use a special type of spline called Extended Cross-Splines or Extended X-Splines (Blanc & Schlick, 1995), for short. In this model, each control point *i* has the coordinates (*xi*, *yi*) and an additional parameter *si* ∈ [0, 1] that allows the curve *C*(*t*) approximates (0 *< si* ≤ 1) or sharp interpolates (*si* = 0) the control point. This feature is very interesting because it allows the definition of smooth or sharp curves. In this work, just smooth curves are represented.

Considering an Extended X-Spline model in which each segment depends on four control points, a segment *C*(*t*) on the parameter range (*tk*+1, *tk*<sup>+</sup>2) is defined as follows:

$$\mathbf{C}(t) = F\_0 \mathbf{P}\_k + F\_1 \mathbf{P}\_{k+1} + F\_2 \mathbf{P}\_{k+2} + F\_3 \mathbf{P}\_{k+3},\tag{2}$$

where the blending functions *Fk* are obtained by the normalization of the blending functions *Ak*(*t*) as follows:

$$F\_k(t) = \frac{A\_k(t)}{A\_0(t) + A\_1(t) + A\_2(t) + A\_3(t)}\tag{3}$$

and their non null part can be divided in two parts, *F*− *<sup>k</sup>* , defined between *T*<sup>−</sup> *<sup>k</sup>* and *Tk*, and *<sup>F</sup>*<sup>+</sup> *k* , defined between *Tk* and *<sup>T</sup>*<sup>+</sup> *k* .

The functions *Ak*(*t*) are obtained by the generic quintic polinomial *f <sup>p</sup>*(*u*), whose coefficients were determined in order to satisfy the constraints of the model, resulting the following expression:

$$f\_p\left(u\right) = u^3 \left(10 - p + \left(2p - 15\right)u + \left(6 - p\right)u^2\right) \,. \tag{4}$$

Hence, for the non null parts of the functions *Ak*(*t*), we have:

$$A\_0(t) = f\_{p\_{k-1}}(\mathfrak{u}\_0) \; , \; A\_1(t) = f\_{p\_k}(\mathfrak{u}\_1) \; , \; A\_2(t) = f\_{p\_{k+1}}(\mathfrak{u}\_2) \; , \; A\_3(t) = f\_{p\_{k+2}}(\mathfrak{u}\_3) \tag{5}$$

with

4 Will-be-set-by-IN-TECH

(a) Systole (b) Diastole

with *ti* = 0 for all *i*, and the one relative to the end of diastole with *ti* = 1 for all *i*. Doing so, the method goal is to recover the shape of the ventricular cavities via the estimation of the 15

Splines are mathematical models that associate a curve with a set of points named control points. Here we use a special type of spline called Extended Cross-Splines or Extended X-Splines (Blanc & Schlick, 1995), for short. In this model, each control point *i* has the coordinates (*xi*, *yi*) and an additional parameter *si* ∈ [0, 1] that allows the curve *C*(*t*) approximates (0 *< si* ≤ 1) or sharp interpolates (*si* = 0) the control point. This feature is very interesting because it allows the definition of smooth or sharp curves. In this work, just

Considering an Extended X-Spline model in which each segment depends on four control

where the blending functions *Fk* are obtained by the normalization of the blending functions

The functions *Ak*(*t*) are obtained by the generic quintic polinomial *f <sup>p</sup>*(*u*), whose coefficients were determined in order to satisfy the constraints of the model, resulting the following

<sup>10</sup> <sup>−</sup> *<sup>p</sup>* <sup>+</sup> (2*<sup>p</sup>* <sup>−</sup> <sup>15</sup>) *<sup>u</sup>* <sup>+</sup> (<sup>6</sup> <sup>−</sup> *<sup>p</sup>*) *<sup>u</sup>*<sup>2</sup>

*A*0(*t*) = *fpk*−<sup>1</sup> (*u*0) , *A*1(*t*) = *fpk* (*u*1) , *A*2(*t*) = *fpk*<sup>+</sup><sup>1</sup> (*u*2) , *A*3(*t*) = *fpk*<sup>+</sup><sup>2</sup> (*u*3) (5)

**C**(*t*) = *F*0**P***<sup>k</sup>* + *F*1**P***k*+<sup>1</sup> + *F*2**P***k*+<sup>2</sup> + *F*3**P***k*+<sup>3</sup> , (2)

*<sup>A</sup>*0(*t*) + *<sup>A</sup>*1(*t*) + *<sup>A</sup>*2(*t*) + *<sup>A</sup>*3(*t*) (3)

*<sup>k</sup>* , defined between *T*<sup>−</sup>

*<sup>k</sup>* and *Tk*, and *<sup>F</sup>*<sup>+</sup>

. (4)

*k* ,

points, a segment *C*(*t*) on the parameter range (*tk*+1, *tk*<sup>+</sup>2) is defined as follows:

*Fk*(*t*) = *Ak*(*t*)

and their non null part can be divided in two parts, *F*−

*k* .

*fp* (*u*) = *u*<sup>3</sup>

Hence, for the non null parts of the functions *Ak*(*t*), we have:

Fig. 2. The control points are represented by red crosses. The dashed lines are the search

space.

**2.1.2 Splines**

*Ak*(*t*) as follows:

expression:

defined between *Tk* and *<sup>T</sup>*<sup>+</sup>

parameters *ti*, with *i* = 1, 2, ..., 15.

smooth curves are represented.

$$\mu\_0 = \frac{t - T\_k^+}{t\_k - T\_k^+}, \; \mu\_1 = \frac{t - T\_{k+1}^+}{t\_{k+1} - T\_{k+1}^+}, \; \mu\_2 = \frac{t - T\_{k+2}^-}{t\_{k+2} - T\_{k+2}^-}, \; \mu\_3 = \frac{t - T\_{k+3}^-}{t\_{k+3} - T\_{k+3}^-} \tag{6}$$

and

$$p\_{k-1} = 2\left(t\_k - T\_k^+\right)^2 \; \; \; p\_k = 2\left(t\_{k+1} - T\_{k+1}^+\right)^2 \; \; \; \tag{7}$$

$$p\_{k+1} = 2\left(t\_{k+2} - T\_{k+2}^{-}\right)^2 \ , \ p\_{k+2} = 2\left(t\_{k+3} - T\_{k+3}^{-}\right)^2 \ . \tag{8}$$

The flexibility of the model is improved by the introduction of new degree of freedom *sk*. In each control point **<sup>P</sup>***<sup>k</sup>* this new parameter affects the values *<sup>T</sup>*<sup>+</sup> *<sup>k</sup>*−<sup>1</sup> <sup>e</sup> *<sup>T</sup>*<sup>−</sup> *<sup>k</sup>*+<sup>1</sup> as follows:

$$T\_{k-1}^{+} = t\_k + s\_{k \text{ \textquotedblleft}} \ T\_{k+1}^{-} = t\_k - s\_k \tag{9}$$

In other words, one can say that the parameter *s* controls the distance between the curve and the control point, what allows the approximation or the interpolation of these points, as shown in Fig. 3.

Fig. 3. The values of the parameters of each control point are: *s*<sup>0</sup> = 0, *s*<sup>1</sup> = 1, *s*<sup>2</sup> = 0, *s*<sup>3</sup> = 0, *s*<sup>4</sup> = 0, *s*<sup>5</sup> = 1, *s*<sup>6</sup> = 0. These figures were inspired by the work of Blanc & Schlick (1995).

Although the expressions used to implement the Extended X-spline were rewritten above, detailed description about this and other types of splines can be found in Blanc & Schlick (1995).

#### **2.1.3 Resistivity model**

The proposed 2D model splits the domain in regions that represent different biological tissues, heart cavities, lungs and torso, each one associated with a different electrical resistivity.

Grimnes (2008) presents the main factors that influence the properties of biological tissues. Although they may be classified in only four groups ( epithelium, muscle, connective tissue and nervous tissue), the tissues can be divided in thirty kinds in accordance to their electrical properties (Gabriel, 1996). In addition, the value of the resistivity of each tissue depends on the frequency of the electrical current, on the temperature, on the presence of water, among other issues.

In this work, we assume the resistivity of each tissue as known, constant and isotropic. These are all simplified assumptions, since biological tissues are usually heterogeneous and

and these compatibility equations:

Γ*ie*

*ρL*∇*u* = *ρT*∇*u* , **x** ∈ Γ<sup>1</sup> (13)

*ρB*∇*u* = *ρT*∇*u* , **x** ∈ Γ<sup>2</sup> (14)

where Ω = Ω*<sup>L</sup>* + Ω*<sup>B</sup>* + Ω*T*, Γ<sup>1</sup> is the interface between the lung and torso region, Γ<sup>2</sup> is the interface between the blood and the torso region, Γ<sup>3</sup> is the external boundary of the thorax,

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 259

<sup>3</sup> is the part of Γ<sup>3</sup> where the ith electrode is, *Ji* is the electrical current injected on the i-*th* electrode and *ρL*, *ρ<sup>B</sup>* and *ρ<sup>T</sup>* are the resistivities of the lung, blood and torso, respectively.

Fig. 4. The simplified thorax model. Here, the electrodes are represented in green. The

*<sup>p</sup>*∗(*ξ*; **<sup>x</sup>**)*u*(*ξ*; **<sup>x</sup>**)*d*Γ(**x**) =

In order to solve Equation 10 for each subregion the Boundary Element Method (BEM) is used. Further details about this technique can be found in Brebbia et al. (1984). The integral

where *ξ* is the collocation point, Γ is the boundary of the sub-domain, *u* is the electrical potential, *p* is its derivative, *u*∗ and *p*∗ are the fundamental solutions for the potential and its normal derivative, respectively, and *c*(*ξ*) is a function of the boundary shape, whose value is 0 if *ξ* is outside of the domain, 1 if *ξ* ∈ Ω and *β*/2*π* if *ξ* ∈ Γ. The parameter *β* is the angle

To obtain a numerical solution for Equation 15, the boundary of the body is discretized. The external boundary is divided in *N*<sup>0</sup> elements and each subregion boundary in *Nk* elements. In this work, the element adopted approximates the geometry linearly and the value of the electrical potential is considered constant in each element. In this case, the parameter *β* = *π* and then *c*(*ξ*) = 0.5 if *ξ* ∈ Γ. Each boundary element has two nodes for the geometrical definition and a centered node, called functional node, where the potential and

Γ

*u*∗(*ξ*; **x**)*p*(*ξ*; **x**)*d*Γ(**x**) , (15)

regions **L** represent the lungs, **B** the blood and **T** the torso.

Γ

between the left and right tangents at the collocation point *ξ*.

**2.2.1 Numerical solution of Laplace's equation**

*<sup>c</sup>*(*ξ*)*u*(*ξ*) +

equation of BEM for this problem is

anisotropic. However, biological tissues are difficult to characterize, and the reported values vary substantially in the literature. Table 1 presents some resistivity values for blood, heart and lung found in the literature.



For the remaining tissues that compose the section of the thorax, called here torso region, Bruder et al. (1994) proposes a mean resistivity of 500Ω*cm*. The resistivity of the air is 1020Ω*cm*, but the resistivity of the lung filled of air is difficult to determine. Rush et al. (1963) presents a very simplified resistivity distribution model of the thorax characterized by the presence of cavities filled of blood, surrounded by homogeneous material with resistivity ten times greater. The same scheme, properly extended to include the lung regions, is used in this work. Preliminarily, the resistivity of the blood is here taken as 100Ω*cm* and the torso to be 1000Ω*cm*. Two different values were tested for the resistivity of the lungs: 20000Ω*cm* (Ratio of Lung to Torso resistivity, RLT = 20) and 50000Ω*cm* (RLT = 50).

#### **2.2 Forward problem**

The forward problem consists on calculating the electrical potential on the external boundary of the torso that is generated by the current injection on a pair of electrodes. Figure 4 presents the simplified model of the thorax.

Given that our 2D model has three regions with different but constant and isotropic conductivities (heart cavities full of blood, Ω*B*, lungs, Ω*L*, and torso, Ω*T*) the electrical potential *u* at each point of the regions must satisfy Laplaces' equation:

$$
\nabla^2 \mu(\mathbf{x}) = 0 \; \; \; \; \mathbf{x} \in \Omega \; \; \; \tag{10}
$$

the following boundary conditions:

$$\frac{1}{\rho\_T} \frac{\partial u}{\partial \mathbf{n}} = f\_{\text{i}} \quad \mathbf{x} \in \Gamma\_3^{i\varepsilon} \tag{11}$$

$$\frac{\partial \mu}{\partial \mathbf{n}} = \mathbf{0} \,, \quad \mathbf{x} \in \left(\Gamma\_3 - \Gamma\_3^{ie}\right) \tag{12}$$

and these compatibility equations:

6 Will-be-set-by-IN-TECH

anisotropic. However, biological tissues are difficult to characterize, and the reported values vary substantially in the literature. Table 1 presents some resistivity values for blood, heart

150 (Barber & Brown, 1984)

100 (Schwan & Kay, 1956) 400 (Patterson & Zhang, 2003)

400 - 800 (Baysal & Eyuboglu, 2000) 727 - 2363 (Barber & Brown, 1984)

600 - 2000 (Baysal & Eyuboglu, 2000)

<sup>∇</sup>2*u*(**x**) = 0 , **<sup>x</sup>** <sup>∈</sup> <sup>Ω</sup> , (10)

<sup>3</sup> (11)

<sup>3</sup> ) (12)

**Tissue Resistivity** (Ω*cm*)

Blood 150 (Yang & Patterson, 2007)

Heart 250 (Yang & Patterson, 2007)

Lung 1400 (Patterson & Zhang, 2003)

For the remaining tissues that compose the section of the thorax, called here torso region, Bruder et al. (1994) proposes a mean resistivity of 500Ω*cm*. The resistivity of the air is 1020Ω*cm*, but the resistivity of the lung filled of air is difficult to determine. Rush et al. (1963) presents a very simplified resistivity distribution model of the thorax characterized by the presence of cavities filled of blood, surrounded by homogeneous material with resistivity ten times greater. The same scheme, properly extended to include the lung regions, is used in this work. Preliminarily, the resistivity of the blood is here taken as 100Ω*cm* and the torso to be 1000Ω*cm*. Two different values were tested for the resistivity of the lungs: 20000Ω*cm* (Ratio

The forward problem consists on calculating the electrical potential on the external boundary of the torso that is generated by the current injection on a pair of electrodes. Figure 4 presents

Given that our 2D model has three regions with different but constant and isotropic conductivities (heart cavities full of blood, Ω*B*, lungs, Ω*L*, and torso, Ω*T*) the electrical

*<sup>∂</sup>***<sup>n</sup>** <sup>=</sup> *Ji* , **<sup>x</sup>** <sup>∈</sup> <sup>Γ</sup>*ie*

*<sup>∂</sup>***<sup>n</sup>** <sup>=</sup> 0 , **<sup>x</sup>** <sup>∈</sup> (Γ<sup>3</sup> <sup>−</sup> <sup>Γ</sup>*ie*

Table 1. Resistivity values of biological tissues that are found in the literature.

of Lung to Torso resistivity, RLT = 20) and 50000Ω*cm* (RLT = 50).

potential *u* at each point of the regions must satisfy Laplaces' equation:

1 *ρT ∂u*

*∂u*

and lung found in the literature.

**2.2 Forward problem**

the simplified model of the thorax.

the following boundary conditions:

$$
\rho\_L \nabla \mu = \rho\_T \nabla \mu \,, \quad \mathbf{x} \in \Gamma\_1 \tag{13}
$$

$$
\rho\_B \nabla \mu = \rho\_T \nabla \mu \quad \text{ } \mathbf{x} \in \Gamma\_2 \tag{14}
$$

where Ω = Ω*<sup>L</sup>* + Ω*<sup>B</sup>* + Ω*T*, Γ<sup>1</sup> is the interface between the lung and torso region, Γ<sup>2</sup> is the interface between the blood and the torso region, Γ<sup>3</sup> is the external boundary of the thorax, Γ*ie* <sup>3</sup> is the part of Γ<sup>3</sup> where the ith electrode is, *Ji* is the electrical current injected on the i-*th* electrode and *ρL*, *ρ<sup>B</sup>* and *ρ<sup>T</sup>* are the resistivities of the lung, blood and torso, respectively.

Fig. 4. The simplified thorax model. Here, the electrodes are represented in green. The regions **L** represent the lungs, **B** the blood and **T** the torso.

#### **2.2.1 Numerical solution of Laplace's equation**

In order to solve Equation 10 for each subregion the Boundary Element Method (BEM) is used. Further details about this technique can be found in Brebbia et al. (1984). The integral equation of BEM for this problem is

$$
\omega(\mathfrak{f})u(\mathfrak{f}) + \int\_{\Gamma} p^\*(\mathfrak{f}; \mathbf{x}) u(\mathfrak{f}; \mathbf{x}) d\Gamma(\mathbf{x}) = \int\_{\Gamma} u^\*(\mathfrak{f}; \mathbf{x}) p(\mathfrak{f}; \mathbf{x}) d\Gamma(\mathbf{x})\,,\tag{15}
$$

where *ξ* is the collocation point, Γ is the boundary of the sub-domain, *u* is the electrical potential, *p* is its derivative, *u*∗ and *p*∗ are the fundamental solutions for the potential and its normal derivative, respectively, and *c*(*ξ*) is a function of the boundary shape, whose value is 0 if *ξ* is outside of the domain, 1 if *ξ* ∈ Ω and *β*/2*π* if *ξ* ∈ Γ. The parameter *β* is the angle between the left and right tangents at the collocation point *ξ*.

To obtain a numerical solution for Equation 15, the boundary of the body is discretized. The external boundary is divided in *N*<sup>0</sup> elements and each subregion boundary in *Nk* elements. In this work, the element adopted approximates the geometry linearly and the value of the electrical potential is considered constant in each element. In this case, the parameter *β* = *π* and then *c*(*ξ*) = 0.5 if *ξ* ∈ Γ. Each boundary element has two nodes for the geometrical definition and a centered node, called functional node, where the potential and

before. In other words, the goal is to estimate the parameter vector **t** that minimizes *f* :

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 261

where *<sup>f</sup>* is the objective function (*<sup>f</sup>* : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>**), **<sup>R</sup>**(**t**) is the residual function (**<sup>R</sup>** : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>***m*), *m* is the number of measures and *n* is the number of parameters. The number of measures

Equation 19 shows that the problem leads to a non-linear least square problem. So, many techniques can be used to solve it. The so called global convergent methods, for example, Genetic Algorithms (Eiben & Smith, 2003; Michalewicz, 1996), has the advantage of avoid the convergence to local minimum. However, they demand a large number of evaluations of the objective function. On the other hand, the local minimization methods converge faster to the minimum because they use the derivatives of the objective function with respect to the parameters. They also demand a suitable initial approximation to converge to the global minimum. Hybrid strategies, that combine the advantages of local and global methods can be used with success (Hsiao et al., 2001; Peters et al., 2010). In this work, the features of the problem allows the use of a local strategy, the called Levenberg-Marquard Method, that will

The Levenberg-Marquardt Method was adopted to solve the non-linear least square problem represented by Equation 19. The detailed description about this method is vastly found in the literature (Dennis & Schnabel, 1996; Fletcher, 1980; Madsen et al., 2004). The Levenberg-Marquardt Method can be understood as the Gauss-Newton method modified by the model trust region approach. In this method, the minimizer of the non-linear least-square problem is obtained iteratively. Each update of **t** is given by the minimizer **t**+ of the following

where **t**<sup>0</sup> is the current value for the vector of minimization parameters and **t**+ is the updated vector. **<sup>R</sup>** is the same residual function mentioned before. **<sup>J</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* is the Jacobian matrix, storing the derivatives of each element of the residual vector with respect to the optimization

The solution of this constrained minimization problem is the updated vector of variables **t**+:

*<sup>T</sup>* **<sup>J</sup>**(**t**0) + *<sup>μ</sup>***<sup>I</sup>**

where **I** is the identity matrix and *μ* is the parameter that modifies the Gauss-Newton method.

−<sup>1</sup>

≤ *δ*0, *μ* = 0, otherwise, *μ* �= 0.

**J**(**t**0)

*<sup>T</sup>* **<sup>R</sup>** (**t**0) , (23)

variables (*Jij* = *∂Ri*/*∂tj*). *δ*<sup>0</sup> is the initial value of the radius of the trust region.

**t**<sup>+</sup> = **t**<sup>0</sup> −

*<sup>T</sup>* **<sup>R</sup>** (**t**0) 2

 **J**(**t**0)

minimizes �**R**(**t**0) + **J**(**t**0)(**t**<sup>+</sup> − **t**0)�<sup>2</sup> (21) subject to �**t**<sup>+</sup> − **t**0�<sup>2</sup> ≤ *δ*<sup>0</sup> . (22)

**R**(**t**)*T***R**(**t**) (19)

**<sup>R</sup>**(**t**) = **<sup>V</sup>**¯ <sup>−</sup> **<sup>V</sup>**(**t**) (20)

*<sup>f</sup>* <sup>=</sup> <sup>1</sup> 2

depends on the adopted protocol to inject current and measure electrical potential.

with

If **J**(**t**0)

*<sup>T</sup>* **<sup>J</sup>**(**t**0)

−<sup>1</sup>

**J**(**t**0)

be briefly described as follows.

**2.3.1 Levenberg-Marquardt method**

constrained linear least-square problem:

its derivative are computed. Thus, the discretized form of Equation 15 for each subregion *k* allows evaluating the potential at each functional node as follows

$$c(\mathfrak{F}\_i)u(\mathfrak{F}\_i) + \sum\_{j=1}^{N\_k} u\_j \int\_{\Gamma\_j} p^\* d\Gamma\_{\bar{J}} = \sum\_{j=1}^{N\_k} p\_j \int\_{\Gamma\_j} u^\* d\Gamma\_{\bar{J}}.\tag{16}$$

where *uj* and *pj* represent the potential and its normal derivative at the *j*-th functional node. The regular integrals are computed numerically by the usual Gauss Quadrature scheme and the singular ones are computed analytically.

The application of Equation 16 for each sub-domain Ω*k*, in addition to the boundary and compatibility conditions (Equations 11 to 14) for the potential and its normal derivative at the functional nodes of the interface elements at Γ0*k*, yields a linear system of algebraic equations that can be expressed in the matrix form as follows:

**Hu** = **Gp** , (17)

where **u** and **p** are vectors that store the values of potential and its derivative, respectively, at the functional nodes of the boundary elements. The matrices **H** and **G** store the respective computed coefficients.

The number of unknowns is the number of the external boundary elements, in which the potential or the flux is unknown, in addition to the double of interface elements, in which the potential and the flux are unknowns. After collect all of them at the vector **y**, Equation 17 can be rewritten as

$$\mathbf{A}\mathbf{y} = \mathbf{b} \tag{18}$$

where **A** is the matrix of coefficients and **b** is the free vector of the linear system.

After determining the unknowns at the boundary, the values of the electrical potential at the nodes in the center of the electrodes without prescribed values are collected in the vector **V**. Such vector will be compared with the vector of measures **V**¯ during the process of solving the inverse problem.

The implementation of the Boundary Elements Method to solve Laplace's equation was written in Fortran language.

#### **2.3 The inverse problem**

As was said before, the aim of the EIT is to generate an image of the electrical resistivity from measures of electrical potential at the external boundary. This problem can be formulated as a minimization problem in which one wants to find the model of electrical resistivity that minimizes the distance between measured (**V**¯ ) and computed (**V**) potentials. In this work, the objective is to recover the shape of the ventricular cavities under the circumstances explained before. Therefore, the resistivity model is obtained via the estimation of the vector **t** defined in Section 2.1.1. In this case, the vector contains 15 parameters *ti*, with *i* = 1, 2, ..., 15 as described before. In other words, the goal is to estimate the parameter vector **t** that minimizes *f* :

$$f = \frac{1}{2} \mathbf{R}(\mathbf{t})^T \mathbf{R}(\mathbf{t})\tag{19}$$

with

8 Will-be-set-by-IN-TECH

its derivative are computed. Thus, the discretized form of Equation 15 for each subregion *k*

where *uj* and *pj* represent the potential and its normal derivative at the *j*-th functional node. The regular integrals are computed numerically by the usual Gauss Quadrature scheme and

The application of Equation 16 for each sub-domain Ω*k*, in addition to the boundary and compatibility conditions (Equations 11 to 14) for the potential and its normal derivative at the functional nodes of the interface elements at Γ0*k*, yields a linear system of algebraic equations

where **u** and **p** are vectors that store the values of potential and its derivative, respectively, at the functional nodes of the boundary elements. The matrices **H** and **G** store the respective

The number of unknowns is the number of the external boundary elements, in which the potential or the flux is unknown, in addition to the double of interface elements, in which the potential and the flux are unknowns. After collect all of them at the vector **y**, Equation 17 can

After determining the unknowns at the boundary, the values of the electrical potential at the nodes in the center of the electrodes without prescribed values are collected in the vector **V**. Such vector will be compared with the vector of measures **V**¯ during the process of solving the

The implementation of the Boundary Elements Method to solve Laplace's equation was

As was said before, the aim of the EIT is to generate an image of the electrical resistivity from measures of electrical potential at the external boundary. This problem can be formulated as a minimization problem in which one wants to find the model of electrical resistivity that minimizes the distance between measured (**V**¯ ) and computed (**V**) potentials. In this work, the objective is to recover the shape of the ventricular cavities under the circumstances explained before. Therefore, the resistivity model is obtained via the estimation of the vector **t** defined in Section 2.1.1. In this case, the vector contains 15 parameters *ti*, with *i* = 1, 2, ..., 15 as described

where **A** is the matrix of coefficients and **b** is the free vector of the linear system.

*p*∗*d*Γ*<sup>J</sup>* =

*Nk* ∑ *j*=1 *pj* Γ*j*

*u*∗*d*Γ*<sup>J</sup>* , (16)

**Hu** = **Gp** , (17)

**Ay** = **b** , (18)

allows evaluating the potential at each functional node as follows

*Nk* ∑ *j*=1 *uj* Γ*j*

*c*(*ξi*)*u*(*ξi*) +

the singular ones are computed analytically.

computed coefficients.

be rewritten as

inverse problem.

written in Fortran language.

**2.3 The inverse problem**

that can be expressed in the matrix form as follows:

$$\mathbf{R(t)} = \dot{\mathbf{V}} - \mathbf{V(t)}\tag{20}$$

where *<sup>f</sup>* is the objective function (*<sup>f</sup>* : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>**), **<sup>R</sup>**(**t**) is the residual function (**<sup>R</sup>** : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>***m*), *m* is the number of measures and *n* is the number of parameters. The number of measures depends on the adopted protocol to inject current and measure electrical potential.

Equation 19 shows that the problem leads to a non-linear least square problem. So, many techniques can be used to solve it. The so called global convergent methods, for example, Genetic Algorithms (Eiben & Smith, 2003; Michalewicz, 1996), has the advantage of avoid the convergence to local minimum. However, they demand a large number of evaluations of the objective function. On the other hand, the local minimization methods converge faster to the minimum because they use the derivatives of the objective function with respect to the parameters. They also demand a suitable initial approximation to converge to the global minimum. Hybrid strategies, that combine the advantages of local and global methods can be used with success (Hsiao et al., 2001; Peters et al., 2010). In this work, the features of the problem allows the use of a local strategy, the called Levenberg-Marquard Method, that will be briefly described as follows.

#### **2.3.1 Levenberg-Marquardt method**

The Levenberg-Marquardt Method was adopted to solve the non-linear least square problem represented by Equation 19. The detailed description about this method is vastly found in the literature (Dennis & Schnabel, 1996; Fletcher, 1980; Madsen et al., 2004). The Levenberg-Marquardt Method can be understood as the Gauss-Newton method modified by the model trust region approach. In this method, the minimizer of the non-linear least-square problem is obtained iteratively. Each update of **t** is given by the minimizer **t**+ of the following constrained linear least-square problem:

$$\text{minimizes } \|\mathbf{R}(\mathbf{t}\_0) + \mathbf{J}(\mathbf{t}\_0)(\mathbf{t}\_+ - \mathbf{t}\_0)\|\_2 \tag{21}$$

$$\text{subject to} \quad \|\mathbf{t}\_{+} - \mathbf{t}\_{0}\|\_{2} \le \delta\_{0} \,. \tag{22}$$

where **t**<sup>0</sup> is the current value for the vector of minimization parameters and **t**+ is the updated vector. **<sup>R</sup>** is the same residual function mentioned before. **<sup>J</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* is the Jacobian matrix, storing the derivatives of each element of the residual vector with respect to the optimization variables (*Jij* = *∂Ri*/*∂tj*). *δ*<sup>0</sup> is the initial value of the radius of the trust region.

The solution of this constrained minimization problem is the updated vector of variables **t**+:

$$\mathbf{t}\_{+} = \mathbf{t}\_{0} - \left(\mathbf{J}\left(\mathbf{t}\_{0}\right)^{T}\mathbf{J}\left(\mathbf{t}\_{0}\right) + \mu\mathbf{I}\right)^{-1}\mathbf{J}\left(\mathbf{t}\_{0}\right)^{T}\mathbf{R}\left(\mathbf{t}\_{0}\right) \text{ \,\,\,\,\,\tag{23}$$

where **I** is the identity matrix and *μ* is the parameter that modifies the Gauss-Newton method. If **J**(**t**0) *<sup>T</sup>* **<sup>J</sup>**(**t**0) −<sup>1</sup> **J**(**t**0) *<sup>T</sup>* **<sup>R</sup>** (**t**0) 2 ≤ *δ*0, *μ* = 0, otherwise, *μ* �= 0.

(a) Diametrical Pattern (b) Alternative Pattern

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 263

Fig. 5. Two stimulation patterns used in this work. Each double arrow indicates one pair of

From MR images taken at the end of the systole and at the end of the diastole the cardiac ventricles were manually segmented. In this work we are considering an image of just one transversal section of the thorax. Therefore, the problem is treated as 2D and an

Here, the section of the cavities were assumed to be proportional to their volumes, i.e. a cylindrical approximation. So that, in accordance to Equation 1, EF is approximated by:

*EF* <sup>=</sup> *EDA* <sup>−</sup> *ESA*

Therefore, after segmentation, the EF is calculated in accordance to Equation 25. The EF of the left ventricle is 59.24% and the EF of the right ventricle is 29, 95%. These values characterize the initial situation of the heart cicle. From this moment, the EIT can be used to monitor the

Later, a cardiac dysfunction was synthetically generated. In this simulated dysfunction model the end-diastolic volume is the same as in the normal cycle but the end-systolic volume is greater than the normal one. In this pathological situation new cardiac cycle, the EF of the left ventricle is 33.01% and the EF of the right ventricle is 16.19%. These dysfunction values are

As mentioned in Section 2.1, we have also tested the methods considering two different 2D models. Each with a different value for the resistivity of the lungs: 20000Ω*cm* (RLT = 20) and

As was said before, the solution depends on the initial guess provided for the local minimization method. So that, for each of 4 optimization problems (2 stimulus patterns times 2 RLT models) we have tested the optimization method with two different initial guesses. One guess corresponds to the shape of the ventricles at the end of the diastole of normal heart, i.e. *ti* = 1, ∀*i* and the other at the end of the systole for the normal tissue, i.e. *ti* = 0, ∀*i* . The initial

where *EDA* is the end-diastolic area and *ESA* is the end-systolic area.

the target values to be estimated by the here proposed method.

*EDA* (25)

driven electrodes.

approximation of the ejection fraction is needed.

**2.4.2 Problem**

EF.

50000Ω*cm* (RLT = 50).

There are some different implementations of this method with respect to the update of the parameter *μ*. In this work, the implementation of the Levenberg-Marquardt Method is provided by MINPACK-1, a standard package of subroutines written in Fortran language to solve non-linear equations and non-linear least squares problems, that is available at the Netlib repository (*http://www.netlib.org/minpack*). More details about the adopted implementation can be found in Moré et al. (1980). In the numerical experiments presented in this work, the subroutine LMDIF of MINPACK-1 was used. Such subroutine demands only the computation of the residual function **R**(**t**). The jacobian matrix is approximated by finite differences. So, the element *Jij* of the Jacobian matrix is computed as follows:

$$J\_{l\dot{j}} = \frac{\partial R\_{\dot{i}}}{\partial t\_{\dot{j}}} \approx \frac{R\_{\dot{i}}(\mathbf{t} + h\_{\dot{j}}\mathbf{e}\_{\dot{j}}) - R\_{\dot{i}}(\mathbf{t})}{h\_{\dot{j}}} \tag{24}$$

where *hj* is a small finite perturbation at the *j*-th element of the original vector of optimization variables **t** and **e***<sup>j</sup>* is the *j*-th column of the identity matrix.

The value of the parameter *hj* is computed by the product <sup>√</sup>*εtj*, where *<sup>ε</sup>* is a parameter provided by the user. If the machine precision is greater than the computed *hj*, this value is substituted by the machine precision.

#### **2.4 Numerical experiments**

#### **2.4.1 Stimulation patterns**

An important aspect of the EIT problem is the choice of the protocols of current injection and electrical potential measurements. Since the problem is ill-conditioned, the suitable choice can be determinant to the success of the image generation. However, a deep study of the influence of different protocols on the solution of the inverse problem is not the focus of this work. More information on this topic can be found in other works, for instance, Peters & Barra (2010). Here we are limited to compare two patterns of electrical current injection. The first is called diametrical and the second is called alternative. Furthermore, all the experiments were done considering 16 electrodes equally spaced on the external boundary of the torso.

The name of the first pattern, diametrical, comes from an analogy. If the domain were circular, the electrodes used to inject current are diametrically opposed. Although the torso is not circular, the name of the pattern remains. In this pattern, 8 different cases of current injection is taken and 13 measures of potential for each case. So, the diametrical pattern yields 104 measures.

The second pattern, here called alternative, is a tentative of illuminating the region of interest better than other regions. Therefore, in this pattern, the electrodes used to inject current, the driven electrodes, are taken near to the heart. So, 6 cases of current injection with 13 measures each one give a total of 78 measures. Each double arrow of Fig. 5 indicates the driven electrode pair in each case of current injection. In both patterns, measurements on driven electrodes are not considered.

It is important to note that, in this work, the "measured" electrical potential values (**V**¯ ) were also synthetically generated, i.e., also numerically obtained.

(a) Diametrical Pattern (b) Alternative Pattern

Fig. 5. Two stimulation patterns used in this work. Each double arrow indicates one pair of driven electrodes.

#### **2.4.2 Problem**

(24)

10 Will-be-set-by-IN-TECH

There are some different implementations of this method with respect to the update of the parameter *μ*. In this work, the implementation of the Levenberg-Marquardt Method is provided by MINPACK-1, a standard package of subroutines written in Fortran language to solve non-linear equations and non-linear least squares problems, that is available at the Netlib repository (*http://www.netlib.org/minpack*). More details about the adopted implementation can be found in Moré et al. (1980). In the numerical experiments presented in this work, the subroutine LMDIF of MINPACK-1 was used. Such subroutine demands only the computation of the residual function **R**(**t**). The jacobian matrix is approximated by finite

> <sup>≈</sup> *Ri*(**<sup>t</sup>** <sup>+</sup> *hj***e***j*) <sup>−</sup> *Ri*(**t**) *hj*

where *hj* is a small finite perturbation at the *j*-th element of the original vector of optimization

The value of the parameter *hj* is computed by the product <sup>√</sup>*εtj*, where *<sup>ε</sup>* is a parameter provided by the user. If the machine precision is greater than the computed *hj*, this value

An important aspect of the EIT problem is the choice of the protocols of current injection and electrical potential measurements. Since the problem is ill-conditioned, the suitable choice can be determinant to the success of the image generation. However, a deep study of the influence of different protocols on the solution of the inverse problem is not the focus of this work. More information on this topic can be found in other works, for instance, Peters & Barra (2010). Here we are limited to compare two patterns of electrical current injection. The first is called diametrical and the second is called alternative. Furthermore, all the experiments were

done considering 16 electrodes equally spaced on the external boundary of the torso.

The name of the first pattern, diametrical, comes from an analogy. If the domain were circular, the electrodes used to inject current are diametrically opposed. Although the torso is not circular, the name of the pattern remains. In this pattern, 8 different cases of current injection is taken and 13 measures of potential for each case. So, the diametrical pattern yields 104

The second pattern, here called alternative, is a tentative of illuminating the region of interest better than other regions. Therefore, in this pattern, the electrodes used to inject current, the driven electrodes, are taken near to the heart. So, 6 cases of current injection with 13 measures each one give a total of 78 measures. Each double arrow of Fig. 5 indicates the driven electrode pair in each case of current injection. In both patterns, measurements on driven electrodes are

It is important to note that, in this work, the "measured" electrical potential values (**V**¯ ) were

also synthetically generated, i.e., also numerically obtained.

differences. So, the element *Jij* of the Jacobian matrix is computed as follows:

*Jij* <sup>=</sup> *<sup>∂</sup>Ri ∂tj*

variables **t** and **e***<sup>j</sup>* is the *j*-th column of the identity matrix.

is substituted by the machine precision.

**2.4 Numerical experiments 2.4.1 Stimulation patterns**

measures.

not considered.

From MR images taken at the end of the systole and at the end of the diastole the cardiac ventricles were manually segmented. In this work we are considering an image of just one transversal section of the thorax. Therefore, the problem is treated as 2D and an approximation of the ejection fraction is needed.

Here, the section of the cavities were assumed to be proportional to their volumes, i.e. a cylindrical approximation. So that, in accordance to Equation 1, EF is approximated by:

$$EF = \frac{EDA - ESA}{EDA} \tag{25}$$

where *EDA* is the end-diastolic area and *ESA* is the end-systolic area.

Therefore, after segmentation, the EF is calculated in accordance to Equation 25. The EF of the left ventricle is 59.24% and the EF of the right ventricle is 29, 95%. These values characterize the initial situation of the heart cicle. From this moment, the EIT can be used to monitor the EF.

Later, a cardiac dysfunction was synthetically generated. In this simulated dysfunction model the end-diastolic volume is the same as in the normal cycle but the end-systolic volume is greater than the normal one. In this pathological situation new cardiac cycle, the EF of the left ventricle is 33.01% and the EF of the right ventricle is 16.19%. These dysfunction values are the target values to be estimated by the here proposed method.

As mentioned in Section 2.1, we have also tested the methods considering two different 2D models. Each with a different value for the resistivity of the lungs: 20000Ω*cm* (RLT = 20) and 50000Ω*cm* (RLT = 50).

As was said before, the solution depends on the initial guess provided for the local minimization method. So that, for each of 4 optimization problems (2 stimulus patterns times 2 RLT models) we have tested the optimization method with two different initial guesses. One guess corresponds to the shape of the ventricles at the end of the diastole of normal heart, i.e. *ti* = 1, ∀*i* and the other at the end of the systole for the normal tissue, i.e. *ti* = 0, ∀*i* . The initial

Ejection Fraction (%) Initial RLT = 50 RLT = 20 Guess RV LV RV LV Diametrical Pattern *ti* = 0 13.00 34.41 15.32 34.22 *ti* = 1 16.09 32.21 15.80 33.04 Alternative Pattern *ti* = 0 12.97 35.86 20.54 29.94 *ti* = 1 18.72 32.84 20.89 29.40 **Target 16.19 33.01 16.19 33.01**

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 265

Table 2. Values of the ejection fraction estimated for the synthetic cardiac dysfunction for two resistivity models (RLT = 20 and RLT = 50), two different stimulation patterns (diametrical and alternative) and two initial guesses (*ti* = 0 and *ti* = 1). The last row shows the target

> Relative Errors (%) Initial RLT = 50 RLT = 20 Guess RV LV RV LV Diametrical Pattern *ti* = 0 19.70 4.24 5.37 3.67 *ti* = 1 0.62 2.42 2.41 0.09 Alternative Pattern *ti* = 0 19.89 8.63 26.87 9.30 *ti* = 1 15.63 0.51 29.03 10.94

Figure 7 allows a geometrical comparison between the final results and the actual (target) curves. In order to make the visualization easier, these figures show the region of interest defined in Fig. 6 without the curves of the lungs. It is important to emphasize that, to make the comparison fair, the results presented in these figures were obtained with the same initial

The results show that, except in one case, the error of the ejection fraction of the left ventricle is smaller than the right ventricle value. The mean relative error of the eight results of the left

Moreover, except in one case, the diametrical pattern provides results closer to the actual values than the alternative pattern. The diametrical pattern provides a mean relative value of

Table 3. Relative errors of the obtained EF with respect to the target values.

ventricle results is 4.98% while the right ventricle is 14.94%.

4.82% while the mean error of the alternative pattern is 15.10%.

values of the EF.

guess, *ti* = 1, ∀*i*.

guesses and the target can be compared in Fig. 6. Thus, the method was executed a total of 8 times (2 stimulus patterns times 2 RLT models times 2 initial guesses). Each execution computes the parameters **t** of the end of the systole. The areas inside the curves defined by **t** are the *ESA*. These values together with the known *EDA* values, that is supposed to be the same as the initial condition of the heart cycle, are used with Equation 25 to compute the EF. Therefore, each execution yields 2 values: EF of the left and right ventricle.

Fig. 6. A typical target (pink) and the two initial guesses (green) given to the optimization procedure. The first one corresponds to the systole and the second one to the diastole. ROI denotes the region of interest.

#### **3. Results**

Table 2 presents the results of our numerical experiments that aim the EF estimation of the synthetically generated cardiac dysfunction. The columns present the results for the models with different values for the resistivity of the lungs: 20000Ω*cm* (RLT = 20) and 50000Ω*cm* (RLT = 50). Each couple of rows presents the comparison of the two stimulation patterns implemented: diametrical and alternative. In addition, for each pair (stimulus pattern, RLT) results are presented for two different initial guesses. The first one corresponds to the shape of the ventricles at the end of the systole for the normal heart, i.e. *ti* = 0, ∀*i* and the other at the end of the diastole of the normal heart, i.e. *ti* = 1, ∀*i*. The last row of the table presents the target values for comparison: 16.19% the EF of right ventricle (RV) and 33.01% the EF of left ventricle (LV).

Table 3 shows the relative errors between the values of EF obtained in each execution and the target values for each ventricle. These values are computed as

$$
\Delta\% = 100 \times \frac{|\tilde{EF} - EF|}{EF} \,\text{\,\,\,}\tag{26}
$$

where Δ% is the error, *EF*˜ is the value achieved by the inverse problem solution for the ejection fraction and *EF* is the target value. The relative errors are used to compute the mean relative errors used to compare patterns, initial guesses and models.

12 Will-be-set-by-IN-TECH

guesses and the target can be compared in Fig. 6. Thus, the method was executed a total of 8 times (2 stimulus patterns times 2 RLT models times 2 initial guesses). Each execution computes the parameters **t** of the end of the systole. The areas inside the curves defined by **t** are the *ESA*. These values together with the known *EDA* values, that is supposed to be the same as the initial condition of the heart cycle, are used with Equation 25 to compute the EF.

> ROI Target Initial Guess

(b) *ti* = 1, ∀*i*

*EF* , (26)

Therefore, each execution yields 2 values: EF of the left and right ventricle.

ROI Target Initial Guess

Fig. 6. A typical target (pink) and the two initial guesses (green) given to the optimization procedure. The first one corresponds to the systole and the second one to the diastole. ROI

Table 2 presents the results of our numerical experiments that aim the EF estimation of the synthetically generated cardiac dysfunction. The columns present the results for the models with different values for the resistivity of the lungs: 20000Ω*cm* (RLT = 20) and 50000Ω*cm* (RLT = 50). Each couple of rows presents the comparison of the two stimulation patterns implemented: diametrical and alternative. In addition, for each pair (stimulus pattern, RLT) results are presented for two different initial guesses. The first one corresponds to the shape of the ventricles at the end of the systole for the normal heart, i.e. *ti* = 0, ∀*i* and the other at the end of the diastole of the normal heart, i.e. *ti* = 1, ∀*i*. The last row of the table presents the target values for comparison: 16.19% the EF of right ventricle (RV) and 33.01% the EF of left

Table 3 shows the relative errors between the values of EF obtained in each execution and the

<sup>Δ</sup>% <sup>=</sup> <sup>100</sup> <sup>×</sup> <sup>|</sup>*EF*˜ <sup>−</sup> *EF*<sup>|</sup>

where Δ% is the error, *EF*˜ is the value achieved by the inverse problem solution for the ejection fraction and *EF* is the target value. The relative errors are used to compute the mean relative

target values for each ventricle. These values are computed as

errors used to compare patterns, initial guesses and models.

(a) *ti* = 0, ∀*i*

denotes the region of interest.

**3. Results**

ventricle (LV).


Table 2. Values of the ejection fraction estimated for the synthetic cardiac dysfunction for two resistivity models (RLT = 20 and RLT = 50), two different stimulation patterns (diametrical and alternative) and two initial guesses (*ti* = 0 and *ti* = 1). The last row shows the target values of the EF.


Table 3. Relative errors of the obtained EF with respect to the target values.

Figure 7 allows a geometrical comparison between the final results and the actual (target) curves. In order to make the visualization easier, these figures show the region of interest defined in Fig. 6 without the curves of the lungs. It is important to emphasize that, to make the comparison fair, the results presented in these figures were obtained with the same initial guess, *ti* = 1, ∀*i*.

The results show that, except in one case, the error of the ejection fraction of the left ventricle is smaller than the right ventricle value. The mean relative error of the eight results of the left ventricle results is 4.98% while the right ventricle is 14.94%.

Moreover, except in one case, the diametrical pattern provides results closer to the actual values than the alternative pattern. The diametrical pattern provides a mean relative value of 4.82% while the mean error of the alternative pattern is 15.10%.

**4. Discussions and conclusions**

the diametrical.

model (Cheney et al., 1999).

**5. Acknowledgements**

PA, USA.

*Physiology* 83(5): 1762–1767.

*E-Scientific Instruments* 17(9): 723–733.

de Janeiro.

**6. References**

The presented results suggest that the proposed methodology allows a suitable indication of the cardiac ejection fraction. We have observed that the error in the ejection fraction predictions for the right ventricle are greater than those found for the left ventricle and this is

Determination of Cardiac Ejection Fraction by Electrical Impedance Tomography 267

Concerning the different patterns for current injection tested in this work, the errors obtained with the diametrical pattern are smaller than those using the alternative pattern, in general. However this fact does not discard the use of the alternative pattern, as it presents good results and spends around 19 min. in a Pentium 4, 3.00 GHz, for a complete solution, 25% less then

Comparing the results obtained with different lung resistivities we may conclude that the inverse problem becomes more difficult to be solved as the RLT increases. Therefore, the results suggest the current injection should be triggered during the expiratory phase, when

The preliminary results presented in this work suggest the proposed technique is a promising diagnostic tool that may offer continuous and non-invasive estimation of cardiac ejection fraction. However, the use of the EIT in real applications demands further improvements. The model could be improved, for instance, by the implementation of the complete electrode

Another point is that, in this work, we assume the resistivities of the tissue known. Future works should include the resistivities of the tissue as parameters of the inverse problem, as well as deeper studies about the electrical properties of biological tissues. Furthermore, the

This work was partially supported by FAPEMIG, UFJF, CAPES, CNPq and FINEP. In particular, the first author would like to thank CAPES for the Master's scholarship at Federal University of Juiz de Fora and CNPq for the Doctoral scholarship at Federal University of Rio

Adler, A., Amyot, R., Guardo, R., Bates, J. H. T. & Berthiaume, Y. (1997). Monitoring changes in

Anderson, E., Bai, Z., Bischof, C., Blackford, L. S., Demmel, J., Dongarra, J. J., Du Croz, J.,

Barber, D. C. & Brown, B. H. (1984). Applied potential tomography, *Journal of Physics*

lung air and liquid volumes with electrical impedance tomography, *Journal of Applied*

Hammarling, S., Greenbaum, A., McKenney, A. & Sorensen, D. (1999). *LAPACK Users' guide (third ed.)*, Society for Industrial and Applied Mathematics, Philadelphia,

behavior of the proposed method when subjected to real data should be evaluated.

in agreement with other techniques, such as with echocardiography.

the air volume and the corresponding lung resistivity are smaller.

Fig. 7. Some results for the diametrical and the alternative pattern and the target.

About the initial guess, both of them provided good results. However, in this experiments, the best results were obtained with the guess of the original diastole curve, *ti* = 1, with a mean relative error of 7.70% against an error of 12.21% for the other initial guess.

The geometrical results presented in Fig. 7, showing only the ventricular cavities, suggest that they become worse in the case the lung resistivity is greater. This behavior is expected because greater resistivities around the region of interest tend to block the electrical current to reach this area. For instance, for the best experimented pattern, the diametrical one, the mean relative error obtained with the greatest resistivity (RLT = 50) is 3.37% while the mean relative error obtained with the other lung resistivity (RLT = 20) is 1.44%.

The best result can be considered the one obtained for RLT = 20, diametrical pattern and *ti* = 1. The relative error in the value of the ejection fraction is 0.09% for the left ventricle and 2.41% for the right ventricle. Figure 7(b) shows this result. In this case it is very difficult to see the difference between the result and the target.

## **4. Discussions and conclusions**

14 Will-be-set-by-IN-TECH

Target Result

Target Result

(d) RLT = 20 - Alternative

(b) RLT = 20 - Diametrical

Target Result

Target Result

Fig. 7. Some results for the diametrical and the alternative pattern and the target.

relative error of 7.70% against an error of 12.21% for the other initial guess.

error obtained with the other lung resistivity (RLT = 20) is 1.44%.

difference between the result and the target.

About the initial guess, both of them provided good results. However, in this experiments, the best results were obtained with the guess of the original diastole curve, *ti* = 1, with a mean

The geometrical results presented in Fig. 7, showing only the ventricular cavities, suggest that they become worse in the case the lung resistivity is greater. This behavior is expected because greater resistivities around the region of interest tend to block the electrical current to reach this area. For instance, for the best experimented pattern, the diametrical one, the mean relative error obtained with the greatest resistivity (RLT = 50) is 3.37% while the mean relative

The best result can be considered the one obtained for RLT = 20, diametrical pattern and *ti* = 1. The relative error in the value of the ejection fraction is 0.09% for the left ventricle and 2.41% for the right ventricle. Figure 7(b) shows this result. In this case it is very difficult to see the

(c) RLT = 50 - Alternative

(a) RLT = 50 - Diametrical

The presented results suggest that the proposed methodology allows a suitable indication of the cardiac ejection fraction. We have observed that the error in the ejection fraction predictions for the right ventricle are greater than those found for the left ventricle and this is in agreement with other techniques, such as with echocardiography.

Concerning the different patterns for current injection tested in this work, the errors obtained with the diametrical pattern are smaller than those using the alternative pattern, in general. However this fact does not discard the use of the alternative pattern, as it presents good results and spends around 19 min. in a Pentium 4, 3.00 GHz, for a complete solution, 25% less then the diametrical.

Comparing the results obtained with different lung resistivities we may conclude that the inverse problem becomes more difficult to be solved as the RLT increases. Therefore, the results suggest the current injection should be triggered during the expiratory phase, when the air volume and the corresponding lung resistivity are smaller.

The preliminary results presented in this work suggest the proposed technique is a promising diagnostic tool that may offer continuous and non-invasive estimation of cardiac ejection fraction. However, the use of the EIT in real applications demands further improvements. The model could be improved, for instance, by the implementation of the complete electrode model (Cheney et al., 1999).

Another point is that, in this work, we assume the resistivities of the tissue known. Future works should include the resistivities of the tissue as parameters of the inverse problem, as well as deeper studies about the electrical properties of biological tissues. Furthermore, the behavior of the proposed method when subjected to real data should be evaluated.

## **5. Acknowledgements**

This work was partially supported by FAPEMIG, UFJF, CAPES, CNPq and FINEP. In particular, the first author would like to thank CAPES for the Master's scholarship at Federal University of Juiz de Fora and CNPq for the Doctoral scholarship at Federal University of Rio de Janeiro.

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**13** 

**Doppler Flowmetry** 

Helena Lenasi

*Slovenia* 

**Assessment of Human Skin Microcirculation** 

Human skin is the largest organ of the body: it accounts for approximately 5% of the total body weight, extends over about 1.8 m2 and has an average thickness of 1-2 mm. Besides providing the mechanical barrier to protect the surface of the body and to prevent water loss it has other important functions. It is engaged in sensory perception and vitamin D metabolism, in inflammation, hemostasis and wound healing. Its role in human

The major role of blood flow to the skin is therefore concerned with temperature regulation. Accordingly, it is not surprisingly that blood flow to the skin goes far beyond its nutritive demands. The nutritive blood flow is estimated to comprise only 20% of the skin blood flow (SkBF), whereas the rest represents the functional blood flow. On the whole, the SkBF has been estimated to amount to 0.3 l/min at rest in thermoneutral conditions. During exposure to cold it can be reduced to almost zero, whereas it can increase up to 8 l/min during strenous exercise in a hot environment. It is thus obvious that it plays an important role in hemodynamics, since during strenous exercise and in severe heat stress it can comprise over 50% of the cardiac output, as compared to only 5% in resting thermoneutral conditions (Johnson, 1996; Rowell, 1993). The range of flows is therefore wide, and, in extreme conditions, high SkBF also represents a burden on the working heart. Indeed, many persons with borderline cardiac failure develop severe failure in a hot weather because of the extra load on the heart and then revert from failure in cool weather. As one of the major functions

of the skin is to eliminate excessive heat, its temperature is generally under 37°C.

In recent years, the cutaneous microcirculation has gained increasing interest. Its easy and noninvasive accessibility renders skin microcirculation an ideal site for measuring. Moreover, as a dynamic structure it may serve as a model for generalized microvascular function as studies have shown a correlation of vascular reactivity between different vascular beds over the body (coronary arteries, brachial artery and skin microcirculation to list a few of them) in health and disease, at least with regard to endothelial function

There is a constant competition between thermoregulatory and non-thermoregulatory challenges. Many different mechanisms, ranging from systemic to local factors, play in

thermoregulation is essential (Roddie, 1983; Rowell, 1993).

**1. Introduction** 

(Holowatz et al., 2008).

**and Its Endothelial Function Using Laser** 

*Institute of Physiology, Medical Faculty, University of Ljubljana* 

Zlochiver, S., Freimark, D., Arad, M., Adunsky, A. & Abboud, S. (2006). Parametric eit for monitoring cardiac stroke volume, Vol. 27, Iop Publishing Ltd, pp. S139–S146.
