### Concept

Localized electric currents can be induced in tissue by propagating ultrasound in the presence of a magnetic field. Consider, for example, an ultrasonic pulse propagating in an electrolytic fluid (such as soft tissue) in the presence of a static magnetic field oriented in a direction perpendicular to the propagation path. The longitudinal particle motion due to the ultrasonic wave moves the ions back and forth through the magnetic field; this results in Lorentz forces on the ions that give rise to an electric current density that oscillates at the ultrasonic frequency. This idea was used as the basis of "Hall effect imaging" proposed by Wen [17] in which Lorentz forces are employed to image the electrical conductivity of tissue. Montalibet et al. also noted that propagating ultrasound in the presence of a magnetic field will generate electric currents in tissue [18]. They also proposed using this effect to measure the electrical conductivity of tissue.

Here we propose a new application: localized stimulation of active (nerve or cortical) tissue by ultrasonically-induced electric fields. Our objective in this paper is to compute the magnitude and the spatial distribution of the these fields. We shall show that analytical solutions for the field distribution can be derived for an ideally collimated ultrasonic beam.

### Theory

Consider an ion in a conductive medium with charge *q*. The longitudinal particle motion of an ultrasonic wave will cause the ion to oscillate back and forth in the medium with velocity **v**. In the presence of a constant magnetic field, **B**
_{0}, the ion is subjected to the Lorentz force

**F** = *q* **v** × **B**
_{0}. (1)

This produces an electric current density given by

**J**
_{0} = (*n*
_{+}
*u*
_{+} + *n*
_{-}
*u*
_{-})**F**, (2)

where *u*
_{+} and *u*
_{-} are the mobilities of the positive and negative ions (assumed to have charges of *q* and -*q*, respectively), and *n*
_{+} and *n*
_{-} are their concentrations. Combining (1) and (2) gives

**J**
_{0} = *q*(*n*
_{+}
*u*
_{+} + *n*
_{-}
*u*
_{-})**v** × **B**
_{0}. (3)

But the electrical conductivity, σ, of the medium is given by

σ = *q*(*n*
_{+}
*u*
_{+} + *n*
_{-}
*u*
_{-}), (4)

so that

**J**
_{0} = σ**v** × **B**
_{0}. (5)

A typical value of the conductivity of tissue is 0.5 Siemens/m. In the most general case, the conductivity σ may be regarded as a complex quantity to account for polarization (or displacement) currents at higher frequencies. From (5), the equivalent electric field is

**E**
_{0} = **v** × **B**
_{0}. (6)

The field, **E**
_{0}, and the current density, **J**
_{0}, oscillate at the ultrasonic frequency in a direction mutually perpendicular to the propagation path (the direction **v**) and the magnetic field **B**
_{0}.

For convenience, we assume harmonic excitation of the ultrasonic wave of the form exp(-*i* ω*t*). Since all our equations are linear, an arbitrary time dependence can be treated by Fourier synthesis. In the following, the time dependence of all quantities, both ultrasonic and electromagnetic, will be exp(-*i* ω*t*), so this factor is dropped. The ultrasonically-induced current density **J**
_{0} can be regarded as an impressed current density that gives rise to scattered (or secondary) electric and magnetic fields **E**
_{
s
}and **B**
_{
s
}that obey the Maxwell's equation

× **B**
_{
s
}= μ_{0}σ**E**
_{
s
}+ μ_{0}
**J**
_{0}, (7)

where μ_{0} is the free-space magnetic permeability. Dielectric properties of tissue can be accounted for by replacing σ everywhere with σ - *i* ωε where ε is the tissue permittivity. At a frequency of about 1 MHz and using typical values for tissue, the quantity ωε will be one to two orders of magnitude less than σ.

Substituting (5) and (6) into (7), we have

× **B**
_{
s
}= μ_{0}σ(**E**
_{
s
}+ **E**
_{0}). (8)

In this equation, the impressed electric field is **E**
_{0}, given by (6), and **E**
_{
s
}is the induced electric field whose source is the Lorentz-induced current density (5). The total electric field is the sum

**E** = **E**
_{
s
}+ **E**
_{0}, (9)

and the total current density is

**J** = σ**E**
_{
s
}+ **J**
_{0} = σ**E**. (10)

Here our objective is to compute the induced field, **E**
_{
s
}, to be substituted into (9) to obtain the total field, **E**.

To solve this problem, consider a collimated ultrasonic beam propagating in the *z*-direction, assumed here to have axial symmetry with a radial profile given by *p*(*r*), where . The particle velocity is then given by:

where *v*
_{0} is the peak particle velocity, is the *z*-directed unit vector along the beam axis, *k*
_{0} = ω/*c*
_{0}, and *c*
_{0} is the ultrasonic wave speed. Although this beam is idealized (e.g., it neglects spreading), (11) is a reasonable approximation to a focused ultrasonic beam in the region of its focus. In the presence of a magnetic field **B**
_{0} = *B*
_{0}
, the resultant Lorentz-induced current density is predicted by (5) to be:

**J**
_{0}(**r**) = *B*
_{0}
*v*
_{0}σ*p*(*r*)
. (12)

When this current density is substituted into (7), one can solve for the induced electric field, **E**
_{
s
}, using standard techniques. As shown in the appendix, the components of **E**
_{
s
}in cylindrical coordinates are found to be

where

and *I*
_{0}(·) and *K*
_{0}(·) are modified Bessel functions of the first and second kind. At this point, for an arbitrary beam profile *p*(*r*), one must resort to a numerical integration of (16). However, the integrals can be evaluated analytically for the special case of an ideally collimated ultrasonic beam whose radial profile is defined by

where *a* is the beam radius. Substituting (17) into (16), we find [19]:

Inserting this result into (13)-(15), we obtain the components of the induced electric field **E**
_{
s
}, as follows. Inside the beam (*r* <*a*), we have

*E*
_{
r
}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*P*
_{
r
}(*r*) sin φ (19)

*E*
_{φ}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*P*
_{φ}(*r*) cos φ (20)

*E*
_{
z
}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*P*
_{
z
}(*r*) sin φ (21)

where

*P*
_{
z
}(*r*) ≡ -*ik*
_{0}
*a K*
_{1}(*k*
_{0}
*a*) *I*
_{1}(*k*
_{0}
*r*), (24)

and outside the beam (*r* >*a*),

*E*
_{
r
}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*Q*
_{
r
}(*r*) sin φ (25)

*E*
_{φ}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*Q*
_{φ}(*r*) cos φ (26)

*E*
_{
z
}(*r*,φ,*z*) = *B*
_{0}
*v*
_{0}
*Q*
_{
z
}(*r*) sin φ (27)

where

*Q*
_{
z
}(*r*) ≡ -*ik*
_{0}
*a I*
_{1}(*k*
_{0}
*a*) *K*
_{1}(*k*
_{0}
*r*). (30)

On the beam boundary (*r* = *a*), the tangential components of the electric field are continuous as expected; that is, *P*
_{φ} (*a*) = *Q*
_{φ} (*a*) and *P*
_{
z
}(*a*) = *Q*
_{
z
}(*a*). This can be checked by setting *r* = *a* in the above equations and using the identity [20]

By comparing *P*
_{
r
}(*a*) and *Q*
_{
r
}(*a*), however, we see that the radial component of the field is discontinuous on the boundary, indicating charge accumulation there. This charge, of course, oscillates at the ultrasonic frequency. The radial electric field discontinuity is seen to be

Δ*E* = *B*
_{0}
*v*
_{0}[*Q*
_{
r
}(*a*) - *P*
_{
r
}(*a*)] sin φ = *B*
_{0}
*v*
_{0} sin φ , (32)

where (31) was used in the last step. This result can be employed to compute the surface charge density, ρ_{
s
}, on the beam boundary by means of the relation ρ_{
s
}= -εΔ*E*, where ε is the tissue dielectric constant.

It can be shown that the radially-dependent terms (22)-(24) and (28)-(30) achieve their maximum values near *r* = *a*, that is, near the beam boundary. In this region, the terms with *k*
_{0}
*r* in the denominator can be neglected compared to the other terms. Thus, the dominate terms when *r* is near *a* are given by:

for *r* <*a*,

*P*
_{
r
}(*r*) = -*B*
_{0}
*v*
_{0}
*k*
_{0}
*a K*
_{1}(*k*
_{0}
*a*) *I*
_{0}(*k*
_{0}
*r*) (33)

*P*
_{φ}(*r*) = 0 (34)

*P*
_{
z
}(*r*) = -*iB*
_{0}
*v*
_{0}
*k*
_{0}
*a K*
_{1}(*k*
_{0}
*a*) *I*
_{1}(*k*
_{0}
*r*), (35)

for *r* >*a*,

*Q*
_{
r
}(*r*) = *B*
_{0}
*v*
_{0}
*k*
_{0}
*a I*
_{1}(*k*
_{0}
*a*) *K*
_{0}(*k*
_{0}
*r*) (36)

*Q*
_{φ}(*r*) = 0 (37)

*Q*
_{
z
}(*r*) = -*iB*
_{0}
*v*
_{0}
*k*
_{0}
*a I*
_{1}(*k*
_{0}
*a*) *K*
_{1}(*k*
_{0}
*r*). (38)

Let us evaluate these functions on the beam boundary (*r* = *a*). The arguments of the Bessel functions are *k*
_{0}
*a* = 2π*a*/λ, so that, for *a* greater than a few wavelengths, we can use the large argument asymptotic approximations [21]:

which gives

where we have defined *E*
_{0} ≡ *B*
_{0}
*v*
_{0} as the magnitude of the impressed electric field, as seen from (6).

Several comments about this solution are worth noting. First, for a beam with a perfectly (and unrealistically) sharp boundary at *r* = *a*, the radial component of the electric-field gradient is infinite on the boundary. This is a mathematical artifact of our solution since the electric field is discontinuous at *r* = *a*. For a real beam, the transition is much more gradual (on the order of an ultrasonic wavelength or more), and charge distribution on the beam boundary will be distributed over this transition region. A second point is that a numerical integration of (16) for a more realistic beam profile, *p*(*r*), that falls off more smoothly than (17), such as a Gaussian profile, results in an electric field smaller in magnitude at the beam boundary than that given by (39) and (40). Thus, the above analytical solution provides a useful, but very qualitative, picture of the field behavior at the beam boundary. It can also be seen from the expressions (22)-(24) that the induced field is near zero at the center of the beam (*r* = 0). In fact, (24) predicts that the *z*-component of the induced field is exactly zero at *r* = 0 and that the other components are very small compared to **E**
_{0} at *r* = 0. The latter statements also hold true for a Gaussian beam profile, as a numerical integration shows. This is a consequence of the fact that **E**
_{
s
}arises essentially due to charge accumulation in the vicinity of the beam boundary. Thus, on the beam axis, the total electric field, **E**
_{0} + **E**
_{
s
}, is close to the Lorentz induced field **E**
_{0}.

We next consider the gradient of the electric field. For activation, we are interested in the rate of change of the component of the electric field along the direction of the axon. If we regard the significant field as **E**
_{0} only, then we note that this field is perpendicular to the beam axis, while the largest component of the gradient points along the beam axis and is given by the derivative of **E**
_{0} with respect to *z*. If the axon is oriented at an angle α with respect to the beam direction, then the component of the electric field along the direction of the axon is *E*
_{0} sin α. Now the rate of change of this component along the axon direction is the derivative of *E*
_{0} sin α with respect to the distance measured along the axon, which is (*ik*
_{0} cos α)*E*
_{0} sin α. The magnitude of this quantity is largest for α = ± 45° and equals *k*
_{0}
*E*
_{0}/2 = *k*
_{0}
*B*
_{0}
*v*
_{0}/2. In the next section, we estimate the size of *E*
_{0} and the gradient *k*
_{0}
*E*
_{0}/2 for a practical example.