Hybrid Inverse Problems and Medical Imaging

Medical Imaging

We refer as medical imaging to a technique of creating images of the human body from minimally invasive procedures. Because of the non intrusive nature of this techniques, medical imaging can be seen as the solution of mathematical inverse problems, were one wants to recover physical properties of a medium using only external measurements. One of the first and simplest example of a medical imaging method is x-ray computed tomography also called computed tomography (CT scan) or computed axial tomography (CAT scan). The main idea of this technique is to use x-rays (i.e., high-frequency electromagnetic waves that propagate approximately in straight lines) to produced tomographic images or ‘sliced-images’ of specific parts of the body. Once an x-ray is send through the body it will get absorbed depending on the radio-density of the tissue. One is interested in recovering the interior radio-density to produce an image of the interior of the body, using only the information of how much the x-ray intensity got absorbed or attenuated throughout the body. [1]

CT_scan

There are many modalities that, instead of using x-rays, use different type of waves to recover different properties of the tissue. For example:

  • Optical tomography – OT: Uses optical waves to recover dielectric permittivity and optical absorption.
  • Electrical impedance tomography – EIT: Uses low-frequency electromagnetic waves to recover electric impedance (conductivity).
  • Elastic tomography: Uses sonic shear waves to recover shear modulus (viscosity).
  • Ultrasound tomography – UT: Uses ultrasound waves to recover bulk compressibility.
  • Single-photon emission computed tomography – SPECT: Uses gamma rays to recover radio tracer distribution.
  • Computed x-ray tomography – CT: Uses x-rays waves to recover radio-density.

Hybrid Inverse Problems

Hybrid inverse problems, sometimes called either coupled-physics inverse problems or multi-wave problems, studies the mathematical framework for medical imaging modalities that combine the best imaging properties of different types of waves (e.g., optical waves, electrical waves, pressure waves, magnetic waves, shear waves, etc).

Single modalities, like OT, EIT, UT, SPECT, MRI (mentioned aboved), focus only in a particular type of wave, and in some setting they can either recover high resolution or high contrast, but not both with the required accuracy. For instance, optical tomography (OT), electrical impedance tomography (EIT) and elastic tomography are high contrast modalities because they can detect small local variations in the electrical and optical properties of a tissue, however because of their low degree of mathematical stability (logarithmic type – usually refer as instability) they are characterized by their low resolution images. On the other hand ultrasound tomography (UT) and magnetic resonance imaging (MRI) are modalities that provide high resolution but not necessarily high enough contrast. For UT the difference between the index of refraction of the healthy and non-healthy tissue is very small to obtain high enough contrast, For MRI the difference rates at which excited atoms, of healthy and non healthy tissue, returned to the equilibrium state again to small.

In some applications of non-invasive medical imaging modalities (e.g., cancer detection) there is a need for high contrast and high resolution images, high contrast contrast discriminates between healthy and non-healthy tissue whereas high resolution is important to detect anomalies at and early stage. Here is an example an ultrasound image that has good contrast but low resolution.

cancer

The aim of hybrid inverse problems is to couple the physics of each particular wave to benefit from their individual imaging advantages. The following table classifies potential couplings that we could have:

Classification of hybrid inverse problems

In there classifications, the physical coupling can be explained by three potential interactions between different waves:

  1. Generation: the interaction of the first wave with the tissue can generate a second kind of wave (photo-acoustic effect or thermo-acoustic effect).
  2. Tagged: the first wave is tagged locally by a the second type of wave.
  3. Movie: the first wave travels much faster than the second type of wave, this difference is used to produce a movie of the slow wave propagation.

Currently hybrid inverse problems is a very active area of research in mathematics. I plan to write a small introduction of most of the hybrid inverse problems that are in the classification above. I will focus in the mathematical formulation but I will give as much physical motivation as possible. Probably the first hybrid inverse problems discovered is photo-acoustic tomography based on the photo-acoustic effect (sound of light). Here  is a simple-interesting article in The Economist that talks precisely about this medical imaging technique.

1. These problem was solved by J. Radon in 1917 for theoretical reasons, 62 years later A. Cormack and G. Hounfield in 1979 rediscover this formula and got the Nobel price in Medicine for their work of x-ray computed tomography (this is an illustrates of the need of interdisciplinary work between mathematics and science).

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What are inverse problems?

Inverse Problems is a research area dealing with inversion of models or data. An inverse problem is a mathematical framework that is used to obtain information about a physical object or system from observed measurements. The solution to this problem is useful because it generally provides information about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied mathematical problems in science and mathematics. There are many different applications including, medical imaging, geophysics, computer vision, astronomy, nondestructive testing, and many others

We illustrate this by considering an example in geophysics, consider a situation where we want to obtain information about the wave speed structure {g} (or in another words information about the metric {g}) inside the earth {\Omega}, from the observed seismic wave-field, denoted by {\Lambda_g}, at the boundary of the earth {\partial \Omega} (the meaning of {\Lambda_g} becomes clear when we give the precise definition in (2), but think of it as the measurements available in the surface of the earth).

geo1

In the previous picture we see the travel seismic waves {g} produced by an earthquake inside the earth, knowledge of the travel seismic waves provide information about the location of oil and minerals deposits. To obtain this information the boundary of the earth is perturbed by an artificial explosion or due to a natural earthquake, this perturbation produces waves that travels through geodesics (trajectories that minimize the travel time of the wave) in the metric {g} and hits the oil and mineral deposits and reflects back to the surface of the earth. We wait up to time {T} until the waves have reached the boundary and we measure the wave-field using a seismograph.

The mathematical formulation of this problem is described by the wave equation in a compact domain {\Omega} as follows:

\displaystyle \begin{array}{rclcr} \partial_t^2u + \triangle_gu &= &0 & \mbox{in}& (0,T)\times M,\\ u(0,x) = \partial_t u(0,x) &=& 0 & \mbox{for}& x\in M,\\ u(t,x) &=& f(t,x)& \mbox{on} & (0,T)\times \partial M, \end{array} \ \ \ \ \ (1)

where {\triangle_g} is the Laplace-Beltrami operator for the metric {g} and {u(x,t)} is the displacement in some direction of the point {x} at time {t}. The information of the seismic wave field is encoded in the hyperbolic Dirichlet to Neumann map

\displaystyle \Lambda_g: f \rightarrow \left. \frac{\partial u}{\partial \nu}\right|_{\partial \Omega} \ \ \ \ \ (2)

where {\nu} is the outer unit co-normal to {\partial \Omega}. Notice that {\Lambda_g} maps the initial perturbation {f} (Dirichlet data) to the recovered wave-field {\frac{\partial u}{\partial \nu}|_{\partial \Omega}} (Neumann data), that carries within information of the inside metric of the earth.

The forward problem of finding {\Lambda_g} from {g} is a well-posed problem under reasonable regularity assumptions over {g}. This means that if we know the underlying geometry of the problem (i.e., the metric {g}) them by solving for {u} the Dirichlet problem (1), with Dirichlet data {f}, we can obtain the Neumann data {\frac{\partial u}{\partial \nu}|_{\partial \Omega}}. We define the forward operator {F} as

\displaystyle F(g) = \Lambda_g,

as said before this is a well-possed operator, but it requires knowledge of the metric {g} in {\Omega} (that is exactly what we are interested in recover).

inverse_problems

In our example the unknown parameter is {g} while the data is {\Lambda_g}. The framework of inverse problems consist in trying to invert this operator and study the properties of this inversion. Basically we reason by asking: If we have enough measurements (i.e., {\Lambda_g = (f,\frac{\partial u}{\partial \nu}|_{\partial \Omega})} for many {f}‘s), what can we say about the metric {g}? If we denote by {F^{-1}} the inverse operator then the central questions that we want to address on inverse problems are:

  • Existence: Given observed data measurements for the system is there any unknown parameters that actually yields this observations?
  • Uniqueness: Can we determine uniquely the unknown parameters by the observed data measurements?
  • Stability: How are the errors in the data measurements amplified in the resulting unknown parameters?
  • Reconstruction: Is there a computationally efficient formula or procedure to recover the unknown parameters from the data measurements?

You can think that the forward problem is mainly concerned in the problem of prediction, in that sense that given all background knowledge of the problem (mathematical model and parameters in the PDE) and initial conditions you can predict what will happen to the solution over time or space. The inverse problem in the other hand is concerned in the problem of recovering information, in the sense that given only some background knowledge of the problem (mathematical model, but not all parameters in PDE) and some data observed in the boundary of your domain, you want to recover the missing information that carries the solution and with it important physical properties that cannot be directly observed

In general this problems are highly non-linear and ill posed and there is not a comprehensive theory that deals with such problems in this generality, so we need to focus on each particular mathematical model one at the time (vaguely speaking, similar models have similar answers). But so, can we recover the the metric and with it the oil and minerals only from measuring the boundary? Actually, there is a partial positive answer to this question that holds even for more general situations, if you are interested you can find my paper (here) that deals with the question of stability for this example.

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