Organ mapping using parelectric spectroscopy - Semantic Scholar
INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 51 (2006) 1623–1631
PHYSICS IN MEDICINE AND BIOLOGY
doi:10.1088/0031-9155/51/6/018
Organ mapping using parelectric spectroscopy T Blaschke1, R Sivaramakrishnan2, M Gross3 and K D Kramer1 1 2 3
Department of Physics, Freie Universit¨at Berlin, Berlin, Germany Department of Physics, Humboldt-Universit¨at zu Berlin, Berlin, Germany Department of Audiology and Phoniatrics, Charit´e University Medicine, Berlin, Germany
Received 18 April 2005, in final form 28 December 2005 Published 1 March 2006 Online at stacks.iop.org/PMB/51/1623 Abstract Whenever physical methods are used in the field of diagnostics, it is necessary to find an unambiguous mapping of the properties of the tested tissues (e.g. normal or pathologic) to their answer to the respective analysis tool such as nuclear magnetic resonance (NMR), ultrasound, x-rays or the relatively new method of parelectric spectroscopy (PS). The well-established non-invasive NMR method has, by now, a sufficiently wide-spread atlas of such mappings. This has to be contrasted to the situation of the PS method where first experiments showed the fulfilment of conditions necessary for any reliable diagnosis, namely the uncertainties of the results being small compared to the differences between the normal and pathologic state of the tissues under test. To help close this gap, we present here results of the behaviour of 12 different organs of mice, taken 20 min after excision and give the dependence of the two most essential PS parameters, the dipole density and the mobility f0 , on the type of healthy organs. To be able to use tumorous tissues preserved in formaldehyde after excision for comparison purposes, we have been measuring the changes of some organs between the fresh state and the preserved state under formaldehyde for over 180 min each.
1. Aims and motivation The basic idea of non-invasively taken data of different tissue types using the NMR technique is to learn about the behaviour of such specimens in a well-defined volume element of the human body. In order to unambiguously judge a possible anomaly or malignancy, a set of T1 and T2 -values must be available in the form of a ‘mapping’ for many types of tissue in all the possible states between normal and pathologic behaviour. For many years such a mapping has existed in the field of NMR diagnostics (Koenig and Brown 2000, Rink 2000, Rakow-Penner et al 2006). The relatively novel PS method cannot, for the time being, rely on such an atlas of tissue characteristics. We are hopeful that the difference in both mobility and density of the polarization answer as yielded by the PS method is sufficiently large between ‘normal’ and ‘anomalous’ due to the following reason: the same quantity that governs the NMR relaxation 0031-9155/06/061623+09$30.00 © 2006 IOP Publishing Ltd Printed in the UK
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rates 1/T1 , 1/T2 is responsible for the dipole mobility f0 , namely the autocorrelation time τC as introduced and interpreted by P Debye. Thus, the ratios of the NMR rates—between 2 and 3 for both types of tissue—should be found in the quantities as measured by parelectric spectroscopy, thus enabling the PS method to be used for diagnostic purposes. The PS method cannot detect deep-seated volume elements; this restriction to the inspection of small volumes near the outer or, when using endoscopic techniques, to the inner surfaces of the human body is compensated, however, by the short measuring times and by a comparably low price of the commercially available frequency analysers necessary for such measurements. Moreover, the two parelectric parameters of interest are being extracted from 400 frequency points sampled in 20 s. This gives a smaller uncertainty in these parameters than the evaluation of the NMR relaxations times T1 , T2 from 4 to 6 echoes or free induction decays. First results from the inspection of anomalies of the vocal cords (Mahlstedt et al 2002) have been so promising that we dared take the next steps into the direction of creating such a mapping using the PS method. Although we have been restricting ourselves to measure 12 different organs of mice (10 different sites in each sample) from 8 animals, the results could be sorted to give a clear dependence of the two PS parameters, density and mobility f0 , of the organs under test. In the next step, we tried—successfully, as we hope—to follow the changes in these parameters from their state shortly after excision to the extrapolated final state when preserving them in formaldehyde. The aim of this latter step is to enable us to use for comparison purposes a wide range of preserved tissues showing malignancies in their final state and all intermediate states. In all organs measured under these conditions, we could find a clear tendency to final values of the changes in and f0 that yield values reasonably near the initial states. 2. Theoretical background Permanent electric dipole moments p tend to align parallel to an external electric field E to yield a polarization P that can easily be measured if the sample under test changes the capacity of a condenser. If the driving field is a radio-frequency field E(ω), the polarization answer consists of a dispersion part P (ω) and an absorption part P (ω). Debye (1930) could describe these answers in terms of the frequency-dependent electric permittivity (ω), as defined by the relation 1 P , (1) (ω) = 1 + 0 E introducing the phenomenological quantity relaxation time τ : , 1 + ω2 τ 2 (2) ωτ σ + . absorption (ω) = 0 ω 1 + ω 2 τ 2 Here, ∞ represents the answer of the induced dipoles, is the density of the permanent dipoles, σ and 0 are the electric conductivity and the influence constant, respectively, and ω is the variable (angular) frequency. Debye gave a physical interpretation of the relaxation time, which by now is known as the Einstein–Debye equation, dispersion
(ω) = ∞ +
4π r 3 η . (3) kT In this picture, the dipole-carrying molecules are assumed to be spheres of radius r that tumble in the electric field, thus suffering a (shear) viscosity η from their surrounding. The quantities τ=
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k and T are Boltzmann’s constant and the (absolute) temperature, respectively. We switch—for mathematical reasons—to the logarithmic variables x = ln(f/MHz) and x0 = ln(f0 /MHz), using the substitution ωτ = 2πf τ = f/f0 . With the abbreviation for the conductivity term σ/0 ω, σ σ = 18 000 e−x , (4) 0 ω S m−1 we can rewrite the above equations in the form that will be used for the extraction of the parameters, density and mobility f0 /MHz, of the permanent electric dipole moments of the sample: , dispersion (x) = ∞ + 1 + e2(x−x0 ) (5) ex−x0 −x , absorption (x) = s e + 1 + e2(x−x0 ) kT . (6) 8π 2 r 3 η To understand the dependence of both parameters, dipole density and dipole mobility f0 /MHz, on the particle density N/V of the dipole-carrying molecules we need two considerations. The definition of permittivity can be written as 1 Ppermanent 1 Pinduced Ppermanent = ∞ + . (7) =1+ + 0 E E 0 E f0 =
mobility
2
From the definition of the average dipole moment, p = p3kTE ,4 and the definition of the orientational polarization, Ppermanent = (N/V )p, we find the relation p2 N → ∼ N/V . (8) 3kT 0 V In the Einstein–Debye relation as given by equation (6) the dependence of f0 on η is much stronger than its dependence on the volume V = 4π r 3 /3 of the dipole-carrying spheres. The dependence of η on the activation energy A can be written in the form of an Arrhenius law, = ∞ + = ∞ +
η(A) = CT e+A/RT ,
(9)
with the gas constant R and constant C. Thus, we have to take into account the strong dependence f0 (A) = C e−A/RT
x0 = ln(f0 /MHz) = −(1/RT )A + C ,
(10)
with constants C and C . In the framework of the theory of rate processes (Eyring et al 1941) the activation energy describes the jump height energy whenever surrounding particles have to give volume for the dipole-carrying molecule to change its position in the electric field. This energy is raised when the particle density N/V is augmented, giving less free space for this necessary kind of tumbling. We have thus a diminishing value for f0 and an augmenting value for when the particle density is raised. In fact, we find—in anticipation of the experimental results—such a pronounced behaviour as described above. All these considerations as based on the validity of Debye’s equations (2) have, however, to be modified as soon as we have to take into account a distribution of activation energies (Kl¨osgen et al 1996) and/or a distribution of masses of the dipole-carrying molecules (Sivaramakrishnan et al 2004, 2005). The algorithms needed for a correct evaluation of the parameters and f0 are well known and have been published elsewhere (Brecht et al 1999). 4
This approximate case of Langevin’s equation holds in our case where pE/kT 1 is guaranteed.
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T Blaschke et al frequency analyser (Hewlett-Packard)
transmitter (synthes.)
notebook (Compaq)
receiver (locked)
IEEE bus steering receiving evaluating
probe
3 mm
sample holder
outer conductor inner conductor dielectric spacing
open-ended coaxial line ‘condensor’ in contact with the samples
Figure 1. A high-frequency electromagnetic wave is fed into an open-ended coaxial cable and the reflected wave is analysed as to its amplitude and phase shift. The cut inner and outer conductors of this coaxial probe form a condenser which in contact with the organic samples changes the reflection behaviour of the set-up. A notebook is used for scanning the frequency in 200 logarithmically equidistant steps in the preset span (0.3, . . . , 300 MHz). From the 200 values of (f ) and the 200 values of (f ) a simple algorithm is able to extract the desired values of and f0 —see the text.
3. Experimental part The central part of the set-up to be applied for parelectric spectroscopy is a commercial frequency analyser (Hewlett-Packard, type 8752 C) suited for use between 300 kHz and 1.3 GHz. An electromagnetic wave of preset frequency is fed into an open-ended coaxial line of real impedance Z0 , the cut inner and outer conductors of which form a condenser ‘filled’ with the sample under test. The condenser with its complex impedance Z(ω) gives rise to a complex reflection coefficient (see figure 1) i Z(ω) − Z0 . (11) , Z(ω) = − ρ= Z(ω) + Z0 ωC0 ( (ω) − i (ω)) This quantity is responsible for a reflected wave that is analysed for its amplitude and phase shift. From the latter two parameters we extract the real and imaginary part, (ω) and (ω), respectively. In order to ‘teach’ the receiver path of the analyser the character of the terminating condenser C0 as given by equation (11), we have to calibrate the entire set-up using a short termination and fill substances of well-known parelectric behaviour, such as water and air. After this calibration procedure, a notebook (Compaq) sweeps the analyser frequency in a preset range applying 200 logarithmically equidistant steps. The analyser output is processed in this notebook to yield the 200 points (ω) and the 200 points of (ω). Using Debye’s equations in their form (5) we have to evaluate the two parameters ∞ and s(σ ) in which we are not interested and the two parameters essential for further discussions, and x0 = ln(f0 /MHz). This procedure has a rather simple algorithm as a basis: the integrals A1 of the first 100 points of the absorption and A2 of the second 100 points give the quantity Z = 10 · A2 − A1 free from s(σ ) and the integrals D1 (D2 ) of the first (second) 100 points yield the quantity N = D1 − D2 free from ∞ . The quotient Q = Z/N is free from and is a monotonous function of x0 that can be calculated in an analytically closed form. From either N(x0 , ) or Z(x0 , ) we finally get the parameter . In the cases where the parameter
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as derived from N = N (x0 , ) and Z = Z(x0 , ) does not have, within the error limits, the same value, we have an underlying distribution of x0 caused by a distribution of masses and/or activation energies of the dipole-carrying molecules. We have, then, to find out the underlying distribution function G(x, x0 , xG ) of half-width xG . This doubly normalized function with the boundary conditions x=+∞ G(x, x0 , xG ) dx = 1, M0 = x=−∞ (12) x=+∞ (x − x0 )G(x, x0 , xG ) dx = 0, M1 = x=−∞
influences via the well-known folding procedure the courses of both (x) and (x). We have three criteria to find out the character of this distribution, namely • the change of slope of (x) in the point of inflection, • the diminished maximum value of (x), and • the larger half-width of (x). 4. Results and discussion The animal experiments had been treated in accordance with the Animal Ethics Committee of the local authorities. In the first experiment 8 healthy adult NMRI mice, 4 male and 4 female aged between 8 and 26 weeks, weighing 25–30 g, were examined. After a short CO2 anaesthesia the mice were sacrificed. Immediately after decapitation the tissue of several organs, brain, liver, lung, heart and kidney, were exposed to parelectric spectroscopy in ten sequences per animal with ten measurements on each organ. In a second experiment 2 healthy adult NMRI mice, 1 male and 1 female aged 10 and 12 weeks, weighing 25–30 g, were examined. Three sequences of parelectric measurements were performed starting with fresh tissue of brain, liver and heart immediately after scarification. The second sequence was carried out after tissue fixation in a 4% formaldehyde solution for 80 min followed by a third one after 160 min of fixation again in a 4% formaldehyde solution. Calibration was done before the first sequence and after three sequences. All average values follow the dependence as predicted in the theoretical part, mainly expressed by equations (8) and (10). The sum of these average values follows the fitting function f0 = f0 (), ln(f0 /MHz) = −0.592 ln() + 6.144.
(13)
One striking difference, however, is the larger error limit in the cases of the ‘inhomogeneous’ measurements (see figure 2) as to be contrasted to the ‘homogeneous’ case. In the first case, the organs are smaller than or comparable to the volume of the electric field before the open-ended coaxial probe. Thus, we get the same couple of f0 - and -values wherever the probe is placed onto the organ. As a consequence, the average rms error is small because of small deviations from the average; therefore we call this case the ‘homogeneous measurement’. In the second case, the organs are relatively large compared to the field-filled volume. Here, the measured values differ depending on the placement of the probe on the organ surface. In this case of ‘inhomogeneous measurement’ we have a wider spread of values around the average value. The error bars as given in figure 2 are the error bars of the average values of both and f0 . To further investigate the influence of the site of the coaxial probe on the organ surface, especially in the case of the ‘inhomogeneous’ types of measurement, we have plotted the dependence f0 = f0 () for each single organ. Figure 3 gives two examples where—in contrast to the
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8.00
7.00
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5.00
4.00
3.00
homogeneous measurements inhomogeneous measurements 2.00 1000
2000
3000
4000
5000
6000
7000
8000
Figure 2. The behaviour of the 12 organs measured is depicted with respect to the structure of the tissue types. As can easily be seen, there is a strong interdependence between the dipole mobility f0 and the dipole density . Each point represents the average of 80 measurements taken from 8 animals. The proof for the existence of such an interdependence is given for two organs as an example in figure 3.
fitting function (13)—a linear approximation is sufficient to describe the behaviour. Here, the large error bars of the average reduce to the strikingly smaller values of the deviation of each pair of the quantities and f0 from the approximate linear dependence! This, in our opinion, is an additional proof for the argumentation which led us to equations (8) and (10). In addition to this first aim, namely to find out whether the sensitivity and spatial resolution (at a given measuring time) are sufficient to yield results reliable to distinguish between the different behaviour of different organs/types of tissue, we had a second motivation: is it possible to follow in reasonable data acquisition times the changes in both parameters and f0 when the freshly excised samples were fixated in formaldehyde solutions? Three examples are depicted in figure 4, namely the changes from initial values in , f0,in to well-defined final values fin , f0,fin after the data have been taken at times 0 min, 80 min and 160 min. All these results have in common that we can give (within the limits of error) functions of the type f0 (t) = f0,fin + (f0,in − f0,fin ) e−t/τ ,
(t) = fin + (in − fin ) e−t/τ .
(14)
The best characterization to answer the above given question is to regard the parameter p as defined for the two quantities measured by pdensity = fin /in > 1 and pmobility = f0,fin /f0,in < 1. For the behaviour of the three organs given in figure 4 we can state that 1.25 pdensity 1.43
and
0.77 pmobility 0.80.
(15)
These results make, in our opinion, fixated material in the normal and the pathologic state accessible for comparison purposes. The model which can explain the increase in
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5.00
4.00
3.00
02/18/05 - two animals
f0/MHz
Heart
500
∆ε 1500
2500
5.00
4.00
02/18/05 - two animals
f0/MHz
Liver 3.00 600
∆ε 1200
1800
Figure 3. Dipole mobility f0 plotted versus dipole density for two organs and two animals, respectively. Without taking into account the interdependence between f0 and we would calculate the large error bars as given by the crosses through the ‘centre of gravity’. The dependence f0 () of a linear form yields the much smaller error region within the two parallel lines.
dipole density in parallel to a decrease in dipole mobility when the organs are submitted to formaldehyde solutions has the following basic idea: the electric dipole moment of formaldehyde is stronger than the electric dipole moment of the tissue water attached to the headgroup dipoles of the phospholipids forming the cell membranes. Thus, the solvation water shell is being replaced step by step by a solvation formaldehyde shell as the latter molecules are subject to a larger attractive force in the headgroup inhomogeneous dipolar field. The larger dipole moment of the latter molecules attached to the headgroup dipoles are, therefore, responsible for the measured increase in the overall dipole density, and their larger mass has to lower the mobility of the combination headgroup plus solvation shell. 5. Perspectives of the method The relatively novel tool of parelectric spectroscopy is a non-invasive method which yields new insights of medical relevance. The experimental set-up is not expensive, easy to handle, and can rely on a set of quick evaluation algorithms. Furthermore, this method can be used under endoscopic boundary conditions, for instance, for diagnostics and/or therapy control in liver tissue. Further research has to verify whether parelectric spectroscopy is able to differentiate reliably between normal and pathologic tissue.
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Figure 4. Both the dipole mobility f0 and the dipole density are presented in dependence of the time t. At time zero the measured points have been taken for the freshly excised organs: heart, liver and brain, respectively. The later results are after 80 min and after 160 min fixation in a 4% formaldehyde solution. The points represent the averages over ten individual measurements for each organ. The error bars as given are valid for all results. The fitting curves of the form y(t) = a +b exp(−c ·t) with y standing for f0 /MHz and yield the initial values yin = y(t = 0), the asymptotic values yfin = y(t → ∞) and the characterizing parameters p = yfin /yin for each organ measured.
Acknowledgments We are indebted to the project FG 463 as sponsored by the Deutsche Forschungsgemeinschaft for the possibility of using the experimental set-up for parelectric spectroscopy. We thank Frau Dr Ana Stiglic, responsible for the Animal Laboratory of our university, for her valuable help.
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References Brecht M, Kl¨osgen B, Reichle C and Kramer K D 1999 Distribution functions in the description of relaxation phenomena Mol. Phys. 96 149–60 Debye P 1930 Polare Molekeln (Leipzig: S Hirzel) Eyring H, Glasstone S and Laidler K J 1941 The Theory of Rate Processes (New York: McGraw-Hill) Kl¨osgen B, Reichle C, Kohlsmann S and Kramer K D 1996 Parelectric spectroscopy as a sensor for membrane headgroup mobility and hydration Biophys. J. 71 3251–60 Koenig S H and Brown R D 2000 Relaxometry of tissue Methods in Biomedical Magnetic Resonance Imaging and Spectroscopy vol 1 (New York: Wiley) Mahlstedt K, Blaschke T, Kramer K D and Gross M 2002 Parelektrische spektroskopie zur nichtinvasiven diagnostik von larynxgewebe Biomed. Tech. 47 70–5 Rakow-Penner R A, Daniel B, Yu H, Sawyer-Glover A and Glover G H 2006 Relaxation times of breast tissue at 1.5T and 3T using IDEAL J. Magn. Reson. Imaging 23 87–91 Rink P R 2000 Relaxation measurements in whole body MRI Methods in Biomedical Magnetic Resonance Imaging and Spectroscopy vol 1 (New York: Wiley) Sivaramakrishnan R, Kankate L, Niehus H and Kramer K D 2005 Parelectric spectroscopy of drug-carrier-systems— distribution of carrier masses or activation energies Biophys. Chem. 114 221–8 Sivaramakrishnan R, Nakamura C, Mehnert W, Korting H C, Kramer K D and Sch¨afer-Korting M 2004 Glucocorticoid entrapment into lipid carriers—characterization by parelectric spectroscopy and influence on dermal uptake J. Control. Release 97 493–502
PHYSICS IN MEDICINE AND BIOLOGY
doi:10.1088/0031-9155/51/6/018
Organ mapping using parelectric spectroscopy T Blaschke1, R Sivaramakrishnan2, M Gross3 and K D Kramer1 1 2 3
Department of Physics, Freie Universit¨at Berlin, Berlin, Germany Department of Physics, Humboldt-Universit¨at zu Berlin, Berlin, Germany Department of Audiology and Phoniatrics, Charit´e University Medicine, Berlin, Germany
Received 18 April 2005, in final form 28 December 2005 Published 1 March 2006 Online at stacks.iop.org/PMB/51/1623 Abstract Whenever physical methods are used in the field of diagnostics, it is necessary to find an unambiguous mapping of the properties of the tested tissues (e.g. normal or pathologic) to their answer to the respective analysis tool such as nuclear magnetic resonance (NMR), ultrasound, x-rays or the relatively new method of parelectric spectroscopy (PS). The well-established non-invasive NMR method has, by now, a sufficiently wide-spread atlas of such mappings. This has to be contrasted to the situation of the PS method where first experiments showed the fulfilment of conditions necessary for any reliable diagnosis, namely the uncertainties of the results being small compared to the differences between the normal and pathologic state of the tissues under test. To help close this gap, we present here results of the behaviour of 12 different organs of mice, taken 20 min after excision and give the dependence of the two most essential PS parameters, the dipole density and the mobility f0 , on the type of healthy organs. To be able to use tumorous tissues preserved in formaldehyde after excision for comparison purposes, we have been measuring the changes of some organs between the fresh state and the preserved state under formaldehyde for over 180 min each.
1. Aims and motivation The basic idea of non-invasively taken data of different tissue types using the NMR technique is to learn about the behaviour of such specimens in a well-defined volume element of the human body. In order to unambiguously judge a possible anomaly or malignancy, a set of T1 and T2 -values must be available in the form of a ‘mapping’ for many types of tissue in all the possible states between normal and pathologic behaviour. For many years such a mapping has existed in the field of NMR diagnostics (Koenig and Brown 2000, Rink 2000, Rakow-Penner et al 2006). The relatively novel PS method cannot, for the time being, rely on such an atlas of tissue characteristics. We are hopeful that the difference in both mobility and density of the polarization answer as yielded by the PS method is sufficiently large between ‘normal’ and ‘anomalous’ due to the following reason: the same quantity that governs the NMR relaxation 0031-9155/06/061623+09$30.00 © 2006 IOP Publishing Ltd Printed in the UK
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rates 1/T1 , 1/T2 is responsible for the dipole mobility f0 , namely the autocorrelation time τC as introduced and interpreted by P Debye. Thus, the ratios of the NMR rates—between 2 and 3 for both types of tissue—should be found in the quantities as measured by parelectric spectroscopy, thus enabling the PS method to be used for diagnostic purposes. The PS method cannot detect deep-seated volume elements; this restriction to the inspection of small volumes near the outer or, when using endoscopic techniques, to the inner surfaces of the human body is compensated, however, by the short measuring times and by a comparably low price of the commercially available frequency analysers necessary for such measurements. Moreover, the two parelectric parameters of interest are being extracted from 400 frequency points sampled in 20 s. This gives a smaller uncertainty in these parameters than the evaluation of the NMR relaxations times T1 , T2 from 4 to 6 echoes or free induction decays. First results from the inspection of anomalies of the vocal cords (Mahlstedt et al 2002) have been so promising that we dared take the next steps into the direction of creating such a mapping using the PS method. Although we have been restricting ourselves to measure 12 different organs of mice (10 different sites in each sample) from 8 animals, the results could be sorted to give a clear dependence of the two PS parameters, density and mobility f0 , of the organs under test. In the next step, we tried—successfully, as we hope—to follow the changes in these parameters from their state shortly after excision to the extrapolated final state when preserving them in formaldehyde. The aim of this latter step is to enable us to use for comparison purposes a wide range of preserved tissues showing malignancies in their final state and all intermediate states. In all organs measured under these conditions, we could find a clear tendency to final values of the changes in and f0 that yield values reasonably near the initial states. 2. Theoretical background Permanent electric dipole moments p tend to align parallel to an external electric field E to yield a polarization P that can easily be measured if the sample under test changes the capacity of a condenser. If the driving field is a radio-frequency field E(ω), the polarization answer consists of a dispersion part P (ω) and an absorption part P (ω). Debye (1930) could describe these answers in terms of the frequency-dependent electric permittivity (ω), as defined by the relation 1 P , (1) (ω) = 1 + 0 E introducing the phenomenological quantity relaxation time τ : , 1 + ω2 τ 2 (2) ωτ σ + . absorption (ω) = 0 ω 1 + ω 2 τ 2 Here, ∞ represents the answer of the induced dipoles, is the density of the permanent dipoles, σ and 0 are the electric conductivity and the influence constant, respectively, and ω is the variable (angular) frequency. Debye gave a physical interpretation of the relaxation time, which by now is known as the Einstein–Debye equation, dispersion
(ω) = ∞ +
4π r 3 η . (3) kT In this picture, the dipole-carrying molecules are assumed to be spheres of radius r that tumble in the electric field, thus suffering a (shear) viscosity η from their surrounding. The quantities τ=
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k and T are Boltzmann’s constant and the (absolute) temperature, respectively. We switch—for mathematical reasons—to the logarithmic variables x = ln(f/MHz) and x0 = ln(f0 /MHz), using the substitution ωτ = 2πf τ = f/f0 . With the abbreviation for the conductivity term σ/0 ω, σ σ = 18 000 e−x , (4) 0 ω S m−1 we can rewrite the above equations in the form that will be used for the extraction of the parameters, density and mobility f0 /MHz, of the permanent electric dipole moments of the sample: , dispersion (x) = ∞ + 1 + e2(x−x0 ) (5) ex−x0 −x , absorption (x) = s e + 1 + e2(x−x0 ) kT . (6) 8π 2 r 3 η To understand the dependence of both parameters, dipole density and dipole mobility f0 /MHz, on the particle density N/V of the dipole-carrying molecules we need two considerations. The definition of permittivity can be written as 1 Ppermanent 1 Pinduced Ppermanent = ∞ + . (7) =1+ + 0 E E 0 E f0 =
mobility
2
From the definition of the average dipole moment, p = p3kTE ,4 and the definition of the orientational polarization, Ppermanent = (N/V )p, we find the relation p2 N → ∼ N/V . (8) 3kT 0 V In the Einstein–Debye relation as given by equation (6) the dependence of f0 on η is much stronger than its dependence on the volume V = 4π r 3 /3 of the dipole-carrying spheres. The dependence of η on the activation energy A can be written in the form of an Arrhenius law, = ∞ + = ∞ +
η(A) = CT e+A/RT ,
(9)
with the gas constant R and constant C. Thus, we have to take into account the strong dependence f0 (A) = C e−A/RT
x0 = ln(f0 /MHz) = −(1/RT )A + C ,
(10)
with constants C and C . In the framework of the theory of rate processes (Eyring et al 1941) the activation energy describes the jump height energy whenever surrounding particles have to give volume for the dipole-carrying molecule to change its position in the electric field. This energy is raised when the particle density N/V is augmented, giving less free space for this necessary kind of tumbling. We have thus a diminishing value for f0 and an augmenting value for when the particle density is raised. In fact, we find—in anticipation of the experimental results—such a pronounced behaviour as described above. All these considerations as based on the validity of Debye’s equations (2) have, however, to be modified as soon as we have to take into account a distribution of activation energies (Kl¨osgen et al 1996) and/or a distribution of masses of the dipole-carrying molecules (Sivaramakrishnan et al 2004, 2005). The algorithms needed for a correct evaluation of the parameters and f0 are well known and have been published elsewhere (Brecht et al 1999). 4
This approximate case of Langevin’s equation holds in our case where pE/kT 1 is guaranteed.
1626
T Blaschke et al frequency analyser (Hewlett-Packard)
transmitter (synthes.)
notebook (Compaq)
receiver (locked)
IEEE bus steering receiving evaluating
probe
3 mm
sample holder
outer conductor inner conductor dielectric spacing
open-ended coaxial line ‘condensor’ in contact with the samples
Figure 1. A high-frequency electromagnetic wave is fed into an open-ended coaxial cable and the reflected wave is analysed as to its amplitude and phase shift. The cut inner and outer conductors of this coaxial probe form a condenser which in contact with the organic samples changes the reflection behaviour of the set-up. A notebook is used for scanning the frequency in 200 logarithmically equidistant steps in the preset span (0.3, . . . , 300 MHz). From the 200 values of (f ) and the 200 values of (f ) a simple algorithm is able to extract the desired values of and f0 —see the text.
3. Experimental part The central part of the set-up to be applied for parelectric spectroscopy is a commercial frequency analyser (Hewlett-Packard, type 8752 C) suited for use between 300 kHz and 1.3 GHz. An electromagnetic wave of preset frequency is fed into an open-ended coaxial line of real impedance Z0 , the cut inner and outer conductors of which form a condenser ‘filled’ with the sample under test. The condenser with its complex impedance Z(ω) gives rise to a complex reflection coefficient (see figure 1) i Z(ω) − Z0 . (11) , Z(ω) = − ρ= Z(ω) + Z0 ωC0 ( (ω) − i (ω)) This quantity is responsible for a reflected wave that is analysed for its amplitude and phase shift. From the latter two parameters we extract the real and imaginary part, (ω) and (ω), respectively. In order to ‘teach’ the receiver path of the analyser the character of the terminating condenser C0 as given by equation (11), we have to calibrate the entire set-up using a short termination and fill substances of well-known parelectric behaviour, such as water and air. After this calibration procedure, a notebook (Compaq) sweeps the analyser frequency in a preset range applying 200 logarithmically equidistant steps. The analyser output is processed in this notebook to yield the 200 points (ω) and the 200 points of (ω). Using Debye’s equations in their form (5) we have to evaluate the two parameters ∞ and s(σ ) in which we are not interested and the two parameters essential for further discussions, and x0 = ln(f0 /MHz). This procedure has a rather simple algorithm as a basis: the integrals A1 of the first 100 points of the absorption and A2 of the second 100 points give the quantity Z = 10 · A2 − A1 free from s(σ ) and the integrals D1 (D2 ) of the first (second) 100 points yield the quantity N = D1 − D2 free from ∞ . The quotient Q = Z/N is free from and is a monotonous function of x0 that can be calculated in an analytically closed form. From either N(x0 , ) or Z(x0 , ) we finally get the parameter . In the cases where the parameter
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as derived from N = N (x0 , ) and Z = Z(x0 , ) does not have, within the error limits, the same value, we have an underlying distribution of x0 caused by a distribution of masses and/or activation energies of the dipole-carrying molecules. We have, then, to find out the underlying distribution function G(x, x0 , xG ) of half-width xG . This doubly normalized function with the boundary conditions x=+∞ G(x, x0 , xG ) dx = 1, M0 = x=−∞ (12) x=+∞ (x − x0 )G(x, x0 , xG ) dx = 0, M1 = x=−∞
influences via the well-known folding procedure the courses of both (x) and (x). We have three criteria to find out the character of this distribution, namely • the change of slope of (x) in the point of inflection, • the diminished maximum value of (x), and • the larger half-width of (x). 4. Results and discussion The animal experiments had been treated in accordance with the Animal Ethics Committee of the local authorities. In the first experiment 8 healthy adult NMRI mice, 4 male and 4 female aged between 8 and 26 weeks, weighing 25–30 g, were examined. After a short CO2 anaesthesia the mice were sacrificed. Immediately after decapitation the tissue of several organs, brain, liver, lung, heart and kidney, were exposed to parelectric spectroscopy in ten sequences per animal with ten measurements on each organ. In a second experiment 2 healthy adult NMRI mice, 1 male and 1 female aged 10 and 12 weeks, weighing 25–30 g, were examined. Three sequences of parelectric measurements were performed starting with fresh tissue of brain, liver and heart immediately after scarification. The second sequence was carried out after tissue fixation in a 4% formaldehyde solution for 80 min followed by a third one after 160 min of fixation again in a 4% formaldehyde solution. Calibration was done before the first sequence and after three sequences. All average values follow the dependence as predicted in the theoretical part, mainly expressed by equations (8) and (10). The sum of these average values follows the fitting function f0 = f0 (), ln(f0 /MHz) = −0.592 ln() + 6.144.
(13)
One striking difference, however, is the larger error limit in the cases of the ‘inhomogeneous’ measurements (see figure 2) as to be contrasted to the ‘homogeneous’ case. In the first case, the organs are smaller than or comparable to the volume of the electric field before the open-ended coaxial probe. Thus, we get the same couple of f0 - and -values wherever the probe is placed onto the organ. As a consequence, the average rms error is small because of small deviations from the average; therefore we call this case the ‘homogeneous measurement’. In the second case, the organs are relatively large compared to the field-filled volume. Here, the measured values differ depending on the placement of the probe on the organ surface. In this case of ‘inhomogeneous measurement’ we have a wider spread of values around the average value. The error bars as given in figure 2 are the error bars of the average values of both and f0 . To further investigate the influence of the site of the coaxial probe on the organ surface, especially in the case of the ‘inhomogeneous’ types of measurement, we have plotted the dependence f0 = f0 () for each single organ. Figure 3 gives two examples where—in contrast to the
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Figure 2. The behaviour of the 12 organs measured is depicted with respect to the structure of the tissue types. As can easily be seen, there is a strong interdependence between the dipole mobility f0 and the dipole density . Each point represents the average of 80 measurements taken from 8 animals. The proof for the existence of such an interdependence is given for two organs as an example in figure 3.
fitting function (13)—a linear approximation is sufficient to describe the behaviour. Here, the large error bars of the average reduce to the strikingly smaller values of the deviation of each pair of the quantities and f0 from the approximate linear dependence! This, in our opinion, is an additional proof for the argumentation which led us to equations (8) and (10). In addition to this first aim, namely to find out whether the sensitivity and spatial resolution (at a given measuring time) are sufficient to yield results reliable to distinguish between the different behaviour of different organs/types of tissue, we had a second motivation: is it possible to follow in reasonable data acquisition times the changes in both parameters and f0 when the freshly excised samples were fixated in formaldehyde solutions? Three examples are depicted in figure 4, namely the changes from initial values in , f0,in to well-defined final values fin , f0,fin after the data have been taken at times 0 min, 80 min and 160 min. All these results have in common that we can give (within the limits of error) functions of the type f0 (t) = f0,fin + (f0,in − f0,fin ) e−t/τ ,
(t) = fin + (in − fin ) e−t/τ .
(14)
The best characterization to answer the above given question is to regard the parameter p as defined for the two quantities measured by pdensity = fin /in > 1 and pmobility = f0,fin /f0,in < 1. For the behaviour of the three organs given in figure 4 we can state that 1.25 pdensity 1.43
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0.77 pmobility 0.80.
(15)
These results make, in our opinion, fixated material in the normal and the pathologic state accessible for comparison purposes. The model which can explain the increase in
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Figure 3. Dipole mobility f0 plotted versus dipole density for two organs and two animals, respectively. Without taking into account the interdependence between f0 and we would calculate the large error bars as given by the crosses through the ‘centre of gravity’. The dependence f0 () of a linear form yields the much smaller error region within the two parallel lines.
dipole density in parallel to a decrease in dipole mobility when the organs are submitted to formaldehyde solutions has the following basic idea: the electric dipole moment of formaldehyde is stronger than the electric dipole moment of the tissue water attached to the headgroup dipoles of the phospholipids forming the cell membranes. Thus, the solvation water shell is being replaced step by step by a solvation formaldehyde shell as the latter molecules are subject to a larger attractive force in the headgroup inhomogeneous dipolar field. The larger dipole moment of the latter molecules attached to the headgroup dipoles are, therefore, responsible for the measured increase in the overall dipole density, and their larger mass has to lower the mobility of the combination headgroup plus solvation shell. 5. Perspectives of the method The relatively novel tool of parelectric spectroscopy is a non-invasive method which yields new insights of medical relevance. The experimental set-up is not expensive, easy to handle, and can rely on a set of quick evaluation algorithms. Furthermore, this method can be used under endoscopic boundary conditions, for instance, for diagnostics and/or therapy control in liver tissue. Further research has to verify whether parelectric spectroscopy is able to differentiate reliably between normal and pathologic tissue.
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Figure 4. Both the dipole mobility f0 and the dipole density are presented in dependence of the time t. At time zero the measured points have been taken for the freshly excised organs: heart, liver and brain, respectively. The later results are after 80 min and after 160 min fixation in a 4% formaldehyde solution. The points represent the averages over ten individual measurements for each organ. The error bars as given are valid for all results. The fitting curves of the form y(t) = a +b exp(−c ·t) with y standing for f0 /MHz and yield the initial values yin = y(t = 0), the asymptotic values yfin = y(t → ∞) and the characterizing parameters p = yfin /yin for each organ measured.
Acknowledgments We are indebted to the project FG 463 as sponsored by the Deutsche Forschungsgemeinschaft for the possibility of using the experimental set-up for parelectric spectroscopy. We thank Frau Dr Ana Stiglic, responsible for the Animal Laboratory of our university, for her valuable help.
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