Many HIFU therapies are carried out with a single element transducer which uses the geometry of the transducer to focus the ultrasound energy to a single spot. In order to target multiple locations, the transducer must be physically moved. The beauty of working with phased arrays is that the array need not move to target multiple locations. Phased arrays have the ability to electronically steer the focus to target multiple locations either sequentially or even simultaneously. The electronic steering allows for targets to be updated on a microsecond time scale.

My adviser was one of the first to propose how phased arrays can be used to target multiple locations simultaneously, original journal paper. I’ll try to give a brief overview of the refocusing process before showing a few examples the refocusing you can accomplish with a phased array.

Before diving in here, let me apologize for the matrix, vector formatting. I used an online Latex editor and things did not turn out super nice but I hope I can explain each variable well enough that things are clear. For the purposes of this discussion let’s assume a 64 element phase array. All the operations here can be extended to arrays of any size.

#### Refocusing Mathematics

The first step in refocusing is to define the forward problem, that is, how does exciting the array with a given driving pattern correspond to acoustic pressures at specific points in the field? We can do this with the equation below

In this equation, u is a 64×1 vector which represents the complex driving pattern on the array. During therapy, each element on the array is excited by a mono-chromatic wave which can be defined by its phase and amplitude. This phase and amplitude can be represented very compactly by a single complex number, where the modulus is equal to the amplitude and the argument is equal to the phase of the wave. The Nth element of vector u represents the excitation on array element N. The matrix H is a matrix of size K x 64, where k is the number of synthesized focal points and 64 is, or course, the number of elements. This matrix represents the propagation from the array to the focal points. The entries of H are calculated from the Green’s function of the elements. The entry H(m,n) represents the Green’s function of element n evaluated a focal spot m. Here we are assuming that the propagation is a linear system, so like any LTI system, the response due to a sinusoidal input can be represented as a scaling and a phase shift (i.e. a single complex number just like the excitation). The vector p represents the complex pressures realized at the k focal points due to the driving pattern u. So to summarize, u is the driving vector which we control by the types of signals presented to the array. The matrix H represents the propagation of the ultrasound energy from the array to the focal points and is dictated by the acoustic medium. The vector p represents the realized pressures at the focal points.

The forward problem is nice, it allows us to see how a given driving pattern will effect the target locations. What would be better is if we could specify our desired focal pressures and then synthesize a driving pattern that would create these pressures. If we look at the dimensions of the matrices involved in the forward problem, we see that for K focal points, we have a K x 64 system matrix. This means that if we have fewer focal targets than elements in the array, we should be able to generate an infinite number of driving patterns that will create the desired focal pressures (i.e. the problem is under-determined). In this case, the question becomes, not can we generate a driving pattern, but which driving pattern do we want to use. What we do is subject the driving pattern to some additional criteria. Typically we try to minimize the function u’Wu, where W is a hermitian-symmetric, positive-definite matrix. We can solve for the general form of this solution using a Lagrangian Multipliers approach and we end up with the generic formula below.

At its simplest form, W can be an identity matrix which just minimizes the total energy in the driving pattern. An interesting formulation is to use another propagation matrix to form W such that W = (Hc’ x Hc), where Hc is a propagation matrix from the array to locations in the field where you wish to avoid depositing energy (e.g. ribs, nerves, etc…).

#### Populating H

For most situations, the elements of H are solved for with the Rayleigh-Sommerfeld solution to the Huygens-Fresnel principle. This formulation, shown below, allows the calculations of the complex pressure at point m due to element n.

In this equation, r_{m} is the observation position vector, r_{n} is the element position vector, E_{n} is the vector normal to the surface of the element.

A 3D model of the array surface is used to facilitate the evaluation of the above equation. The surface of each element is diced into thousands of square segments. This dicing allows the surface integral in the Rayleigh-Sommerfeld equation be broken into the summation of many, smaller surface integrals.

If the dicings are made small enough, most reasonable observation points can be assumed to reside in the far-field of the diced segment. This allows the inner surface integral to be replaced with the far-field radiation pattern of a square element. The size of the dicings in the array model, combined with the distance to most observation points, ensures that most points of interest are well within the main lobe of the radiation pattern. It’s for this reason that the sinc-sinc term normally present in the radiation pattern of a square element is assumed unity and ignored. Using these assumptions, each element in H can be evaluated with the following equation.

The diced model of a 1 MHz array is shown below. The elements are placed on a spherical surface with a 100 mm radius of curvature.

The dicings are spaced about an eighth of a wavelength apart, breaking each element into 2136 dicings across the face of the element.

#### Refocusing Examples

Shown below are two basic refocusing examples. The simulated HIFU pattern was ovelaid on a picture of a 64 element phased array. The first example shows a simple double focus pattern.

The next example shows the same target locations, but in this case the ultrasound energy has been refocused around a potential critical structure prefocally.

It’s actually possible to visualize these double focal spots if we direct the ultrasound energy to the surface of a water tank. The ultrasound will interact with the surface of the water to actually push it up creating a fountain while at the same time nebulizing the water to form water vapor. Below you can see a video where a reflector was used to direct the ultrasound energy to the surface causing both focal points to create a visible, misting fountain.

#### Comparison With Shadowing

Another technique commonly used to avoid depositing acoustic energy on critical structures is known as shadowing. Shadowing uses a geometric optics approach to control energy deposition prefocally. A line is drawn between the array element and the focal point, if a critical structure lies on this line, or within some margin, the element is silenced. This is repeated for all elements, resulting in only those elements which have a direct view of the target emitting a wave.

The appeal of shadowing is its simplicity. There are no matrix inversions required, only projecting a line in 3d space. The problem with shadowing is that ultrasound does not travel in a straight line, it is a wave and will diffract. This cuts both ways, an element which does not directly face a critical structure will still deposit energy on this structure. An element which does directly face a critical structure still contributes some energy to the focus. The synthesis method described above incorporates this and ensures that the wave emitted from each element, no matter where the element lies, adds destructively at critical points, and constructively at the focus. In addition, the synthesis method allows the field to be regularized, greatly reducing potentially chaotic interference patterns at the focus.

The figures below show a comparison between the synthesis method of refocusing and the shadowing method of refocusing. The HIFU field pattern has been overlaid on a synthetic aperture image of the medium (array located below the image). The image is used to identify the size and location of the ribs for refocusing. The first image shows the pattern due to synthesis method of refocusing.

The next image shows the field resulting from a shadowing based approach. The energy incident on the ribs is decreased when compared to no refocusing at all, but the focal pattern has an interference pattern with pre and post focal peaks nearly as large as the focus.

Given the advances in computing power, the additional computational burden of the synthesis method is becoming much less of an issue. On my mid-range desktop computer, over a thousand array patterns can be calculated per second for a 64 element array. The following video demonstrates the ability to refocus in real-time. This video simulates a volumetric ablation scenario. A focal point trajectory is identified on the image allowing for the ablation of a large region.

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