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]


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.


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).


About cmontalto

I am a curious individual with a deep desire of making the world better for those in need.
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