A Bipolar Theory of Living Processes

Investigations in which formula (1) is being applied to different problems of practical and theoretical interest are at present being made in this laboratory. In the first of these studies which will *The theory for the simple case of a diluted suspension of spheres is given in the paper on p. 341. be published shortly, the case of cream will be considered. In the present paper the formula will be applied to the case of blood. The most exact investigations of the conductivity of blood in terms of the volume concentration of the red corpuscles have been made by Stewart? and Fraenckel? Stewart has determined the volume concentration by two independent methods, a colorimetric method and the Hoppe-Seyler chemieal method. Fraenckel has used Bleibtreu's chemical method. The agreement between the results secured by these investigations is very close. However, as has been stated also by Fraenckel, one would expect that Stewart’s methods would give the most exact results and therefore we shall here consider only his measurements. In Table 1 is given a series

of measurements made by Stewart* for the blood of a dog. The volume concentration of the normal blood was determined twice by the colorimetric method and twice by the Hoppe-Seyler method. The results were as follows: 41.16, 40.72, 41.52, and 40.99 per cent, the average being 40.98 per cent. A series of red corpuscle suspensions of varying volume concentration was made by concentration or dilution of the normal blood. The volume concentration for each of these suspensions was determined volumetrically using the value of the volume concentration of the original blood. The volume concentrations are given in Table I under р observed. The ratio of the observed conductivity of the serum to

that of the blood is given under а. In order to employ formula а =. 17% b S Furthermore, k, = 0. The formula therefore reads The values of the volume concentration р calculated from this for- kı k The percentile differences between р observed and р calculated are given under A. The deviations throughout are within the limits of experimental errors. Before concluding, it may perhaps be of interest to apply our formula to some measurements of sand suspensions which have been made by Oker-Blom ? and which often are cited in connection with investigations of the conductivity of blood. Two different kinds of sand were used in these investigations. The suspensions were prepared by adding the sand to a hot salt solution containing 3 per cent gelatin and cooling this mixture to gelatination under continuous rotation. No information is given concerning the form of the sand particles except for the statement that the particles of the sand which were used in the second series of experiments

mula using the observed values of — are given under ¢ calculated. (Table 3) were more nearly spherical than those which were used in the first series (Table 2). Our theory is therefore best applied by using Oker-Blom’s observed values of A and р and calobtained from formula (1) by considering Ka = 0. Our calculated results are given in Tables IT and III, the sand used in the measurements in the two tables corresponding to the two kinds of sand used by Oker-Blom. The calculated values for В are constant within experimental errors. The deviation of the value of В from 1.50 is a measure of the deviation of the sand particles from the spherical form.

Department of Biophysics, Cleveland Clinic Foundation From Physical Review, November, 1924, pp. 575-587 The present theory has been developed as a basis for experiments concerning the electric conductivity and capacity * of colloids and * A suspension is electrically equivalent to a certain resistance in parallel with a certain capacity. We refer to this resistance and to this capacity when we speak of the conductivity and capacity of a suspension.

biological cell suspensions, like blood for instance, which for sometime have been carried out in this laboratory. In this paper we shall consider the electric conductivity of a suspension of homogeneous non-polarizable spheroids; in a second paper, the electric capacity of a suspension of polarizable homogeneous spheroids for a current of low frequency; and in a third, the conductivity and capacity for any frequency, of a suspension of spheroids each consisting of a homogeneous interior surrounded by a thin membrane, the conductivity of which is different from the conductivity of the interior. The last calculation includes the case of a suspension of homogeneous polarizable spheroids. A series of measurements illustrating the practical and theoretical applicability of the theory will be given in separate papers. Preliminary reports of some of the measurements have already been presented at meetings of the American Physical Society.

An investigation of the conductivity of a suspension may serve as a means of determining the conductivity of the single particles of the suspension. Important knowledge of the state of the suspended particles may thus be obtained, as in the case of protected metallic colloids, and especially in the case of suspensions of biological cells the life of which seems to be associated with the existence of a characteristic semi-permeability of the exterior cell wall and of the different interphases of the cell interior. In connection with the latter case we may refer to the researches of Brooks,’ of Crile, Hosmer and Rowland,’ of Osterhout? and of many other investigators on the changes of the electric conductivity of bacterial suspensions and of different plant and animal tissues under varying conditions.

The determination of the conductivity of a suspension may also furnish a method for obtaining the volume concentration of the suspension when the specific conductivity of the disperse phase and of the suspending medium is known. This method is used practically in the determination of the volume concentration of the red corpuscles of blood. It seems probable that it may have many other practical applications, as for instance, the determination of the butter-fat of cream.

Finally, the measurement of the conductivity of a suspension з үү, J. У. Osterhout, Injury, Recovery and Death in Relation to Conductivity and Permeability, Phila., 1922. may lead to a determination of the structural factors of the suspended phase. This is possible because as we shall see later, for a constant volume concentration the conductivity of a suspension is to a certain extent dependent on the form (but not on the size) of the suspended particles. This is especially marked when the suspended particles are non-conductors, the condition which is of the most practical interest. This method may, for instance, find practical application in soil analysis.

The determination of the electric conductivity of a suspension belongs to a very important group of physical problems all of which depend on the solution of Poisson’s equation for a twophase system, including Poisson’s theory of induced magnetism, the Clausius-Mossotti theory for the dielectric constant, the Lorenz- Lorentz theory for the index of refraction, etc. Applied to the case of electric conductivity, the formula to which these theories lead, reads

(Е) +8 ° (Fo) 2 in which k, kı and kə are the specific conductivities of the suspension, the suspending and the suspended mediums respectively ; and р is the volume concentration of the suspended medium. . This formula is true theoretically only for the case of a suspension of spheres. The case of a suspension of infinitely well conducting ellipsoids has been treated theoretically by Poisson * in his theory of magnetic induction and also by Lampa?* in his theory of the dielectric constant of a crystal.

According to its theoretical derivation as presented hitherto, Eq. (1) ean be expected to hold only for dilute suspensions. 'The reason for this is that in the given theoretical treatment the electric field around any suspended particle due to the electric charges on the other particles in the suspension is considered as equal to the average value of this field over the whole space of the suspended medium. For the case of a cubic arrangement of spheres, Lord Rayleigh * has shown that the influence of these charges is repre-

sented correctly in Eq. (1) only to the first approximation. It seems difficult theoretically to decide whether the formula holds ‘strictly for the case of a random distribution of the suspended particles. That it does is supported by the well-known fact that there is usually a very close agreement" between the values of the refractive index of a liquid (or dielectric constant, although the agreement here generally is less close) * as determined experimentally and as calculated from formula (1) by means of the experimental value of the refractive index for its vapor. Confirmation was obtained for volume concentrations up to 15 per cent by Millikan ? from his observations of the dielectric constant of emulsions of water in benzol-chloroform. In the following presentation, therefore, we shall take into account the interaction of the suspended particles by means of the procedure employed in deriving formula (1) ; that is, we shall add to the original field the mean value of the forces due to the charges on the suspended particles throughout the whole space of the suspending medium. The exactness of this method is proven by an experimental verification of the formula which we shall derive; to this end we shall in this paper make use of the extensive experimental data which have been accumulated for the electric conductivity of blood.

We shall consider first the general case of a suspension of homogeneous, non-polarizable ellipsoids. The suspension is assumed to be in an electrolytic cell, the dimensions of which are 1X 1X 1 cm. Тһе potential between the electrodes is V volts; the current 4 amperes. Let us consider a single ellipsoid with half axes a, b, c in this suspension. We shall, for the moment, consider а to be parallel to the direction of the electric force. We shall introduce an orthogonal codrdinate system with its origin in the center O of the ellipsoid and its axes a, у, 2 parallel to a, b and с. We shall designate the electric force at an arbitrary point of the suspending phase as №; as stated above, this is composed of the original electric force V due to the surface charges on the electrodes and of the mean value of the forces due to the surface charges on the suspended particles throughout the whole space of the suspending medium.

For the purpose of determining the surface charges of the ellipsoid we shall introduce confocal coórdinates defining these for a point 2, у, 2 as the three values of 0, A, and у, which satisfy the equation The total potential at a point outside the ellipsoid must be of the form where АА = v (а? +1) (02 А) (c +A), and D and D, are constants which are to be determined by the boundary conditions. The boundary conditions to be fulfilled are (1) For 1= ©, Vest must be equal to the potential of the original field ; this condition is fulfilled since the last term becomes zero for А = х.

At the surface of the ellipsoid, Nest ki = Мат ko, Next and Nine being the normal forces at the two sides of the surface of the ellipsoid. The equation indicates that no accumulation of electricity takes place at the surface. Using п for the direction of the пот: mal to the surface of the ellipsoid we have and a similar equation for Nint. Consequently our equation of condition becomes The values of Ving when b ог c are parallel to the electric field, are derived by replacing a in Eq. (6) by b or c. The value of F is determined by the following equation :

Г Fo drd + Г Fa dz аб = У ext int in which Fs is the component of the electric force parallel to 2 and dS an element of area perpendicular to v; the first integral is taken over the space outside the ellipsoids, the other over the space inside the ellipsoids. We obtain, using equation (6) and writing In order to find the conductivity К of the suspension we shall divide the space between the electrodes into an infinite number of volume elements dS, dz.

kı S dz Л P, dS + № J dz f Fao аб = i= Vk. (8) int tnt ext ext where the first of the two integrals is taken over the space inside the ellipsoids and the other over the space outside the ellipsoids. This equation is evidently equivalent to kf dx ГЕ. dS F (kz — kı) Л dt f F,dS = Vk all all ext ext in which the first double integral is taken over the whole space between the electrodes. We obtain This equation corresponds to the case in which a is parallel to the direction of the original field. The equations which correspond to the cases in which b and c are parallel to the original field are obtained by replacing a by b and c.

The equation corresponding to an arbitrary orientation of the ellipsoids is the sum of three equations like (9) Combining this equation with equation (7) we obtain BOO hk Мир) (hs kı) Introducing this value of F/V into equation (10), we obtain finally а@а=а.5,с We shall now confine ourselves to the case of spheroids, that is b — c. We have the following two integrals in equation (11) to integrate The influence of the geometrical factors of the suspended particles is expressed in equation (13) solely in В. Therefore we conclude from the expression for В that for a constant volume concentration, the conductivity of а suspension is independent of

the size of the suspended particles and also nearly independent of the form of the particles when the difference between the conductivities of the suspended and the suspending phases is not very large. This is especially true for suspensions of prolated spheroids which are less conducting than the suspending medium. Putting ТЕ. (ko/key = 1) a (K/h) B (В/П — 1) — p we obtain from equation (13) the following analogue to equation In this section we shall apply the formula which we have derived to the very accurate and extensive experimental data for the conductivity of blood which have been accumulated by Stewart, Bugarszky,? Fraenckel!'? and others. According to these investigators for a current of low frequency the red corpuscles of blood are perfect insulators so that the ratio of the conductivity of blood to that of its serum is dependent only on the volume concentration of the red corpuscles but independent of the absolute value of the conductivity of the serum. Determinations of this ratio are made for volume concentrations up to 90 per cent. For

the same value of the volume concentration, this ratio is very nearly the same for the blood of man, horse, dog and cow.!? Тһе most exact methods for determining the volume concentration have been used by Stewart and Fraenckel. Stewart has used two ` independent methods, a colorimetric and Hoppe-Seyler’s chemical method. Fraenckel has used Bleibtreu’s chemical method. The agreement between the results of Stewart and those of Fraenckel is very good. However, as has been stated also by Fraenckel, one would expect that Stewart’s methods would give the most exact results. We shall, therefore, restrict ourselves to a comparison of the results of Stewart with those secured by the formula. In Table 1 is given a series of measurements made by Stewart 1% for the blood of a dog. Тһе volume concentration of the normal blood was determined twice by the colorimetrie method and twice by Hoppe-Seylers method. The results were 41.16, 40.72, 41.52 and 40.99 per cent, the average ratio being 40.98 per cent. А series of red corpuscle suspensions of varying volume concentration was made up by concentration or dilution of the normal blood. 'The volume concentration for each of these suspensions was determined volumetrically using the value of the volume concentration of the original blood. These volume concentrations are given in Table I under p (obs.). The ratio of the conductivity of the serum to that of the blood is given under k,/k.

In applying our formula, we take k, = 0, and assume a/b = 1/4.25; consequently 2 = 1.05 and В = — 1.95, and either (15) or (17) may be used to calculate the values of the volume concentration р for the observed values of k,/k. These are given under р (cale.). The deviations р (obs.) — р (cale.) given in the fourth column are probably always within the experimental errors.* It may be of interest to apply our formula also to some measurements of sand suspensions which have been made by Oker-Blom,!5 and are often cited in connection with work on the conductivity of blood. Two different kinds of sand were used. Тһе suspensions were prepared by adding the sand to a hot salt solution containing 3 per cent gelatine and cooling this mixture to gelatination

under continuous rotation. Мо information is given concerning the form of the sand particles except for the statement that the partieles of the sand which were used in the second series of experiments (Table 3) were more nearly spherical than those which were used in the first series (Table 2). Our theory is therefore best applied by using Oker-Blom’s observed values of b;/k and e and caleulating the value of х by means of Eq. (17) since * Тре red corpuscles of mammalians according to the most generally accepted data are biconcave in shape but their dimensions especially in the dog have not been accurately measured. The value 1/4.25 for a/b which secures the best agreement in the present case is in good agreement with the observed values for the dimensions. Determinations of the conductivity of the blood of different animals with simultaneous measurements of the dimensions of the red corpuscles will be made in this laboratory; these investigations will include studies of the blood of different non-mammalian vertebrates in which the red corpuscles are approximately ellipsoids.

К. = 0. Our calculated results are given in Tables 2 and 3, one for each of the two kinds of sand used by Oker-Blom. The calculated values for т in each case are constant within experimental errors. It hardly seems probable, however, that the sand particles could have been as flat as these values for x would indicate; it is more probable that the large values for 2 are due to the fact that the sand particles became more or less completely orientated by the rotation. Eq. (14) with a different value for 2 holds also in this case.

II. THE CAPACITY or A SUSPENSION OF CONDUCTING SPHEROIDS SURROUNDED BY A NoN-CoNDUCTING MEMBRANE FOR A CURRENT оғ Low FREQUENCY ! In the following discussion the specific electric capacity of a disperse system is defined as that capacity which when combined in parallel with a certain resistance electrically balances one centimeter cube of the system. The present discussion applies to the case in which the capacity is due to phenomena at the interphases of the system, especially as in biological tissues, the capacity of which, as is well known, is usually very high. The capacity may be due to either or in part to each of two different causes: (1) a polarization at the interphases, and (2) the presence in the interphases of thin poorly conducting membranes which act as static condensers. The calculation presented here applies to the first case only if the polarization resistance is small compared to the impedance of the polarization capacity. The general case, when they are of the same order of magnitude, will be considered in Part III of this series of papers.

In this presentation we shall derive a formula by which we can determine from the capacity and resistance of a suspension of spheroids and their geometrical form the capacity per unit of surface of the suspended particles. One interesting application of the formula is the calculation of the thickness of the membrane when the capacity is due entirely to the static capacity of membranes on the surface of the suspended particles. As will be shown in the following paper, this condition is probably realized for the case

of blood, and, accordingly, we derive the value of about 3 X (10) * cm. as the thickness of the membrane, which surrounds the red corpuscle, using a probable value for the dielectric constant of the membrane. This method will probably be useful also for the case of many non-biologieal suspensions, such as graphite suspensions, which contain well conducting particles surrounded by poorly conducting films, produced by chemical or adsorptive processes.

The fact that capacity of the type here considered is mainly dependent on the state of the interphases, makes it probable that this may be an important characteristie of the colloid properties of the system ; especially may we expect this to be true for the case of biological systems. Аза matter of fact, investigations which have been carried out in this laboratory have confirmed this belief. It may, for instance, be mentioned that it is found that the capacity of a tumor bears a rather constant relation to its malignancy, this relation appearing so constant that it seems probable that the measurement of the capacity may provide a very practical method for diagnosing the malignancy of a tumor.

We shall first consider the case of a suspension of particles of any form (volume concentration being e, and number of particles per се being n), each particle in which is surrounded by а non-conducting membrane of uniform thickness $, and uniform dielectric constant А. Тһе capacity of this membrane per square centimeter is C, = K/A4nt) · (1005/9) uf. The resistance of the interior of a particle for the frequencies to be considered is supposed to be small compared with the impedance represented by the static capacity of the surface membrane. We shall call the specific conductivity of the suspending medium /, and of the suspension К. If furthermore, F? is the average value of the square of the electric force in the suspending medium, we have

This equation expresses the fact that the heat developed in the suspension by the electric current is equal to the heat developed in a homogeneous conductor with the conductivity k for the same driving potential V. We shall now assume, following thereby in principle the method employed in Part I, that the average value of electrostatic energy which is present at the surface of each suspended particle is obtained by placing the particle in a homogeneous