Visual inspection is the oldest technique available to mankind to assess the quality of biological materials such as fruits, vegetables, meat, fish and live human tissue. Moreover, until recently, visual inspection has been the only technique available to mankind to assess the health status of said biological materials from the outer surface thereof.
Recently, camera based optical imaging has made its entry into the medical field. The power of optical imaging lies in the phenomenon that light penetrates up to centimeters into many biological materials. In most of these materials light is scattered strongly and becomes diffuse. In addition, very specific wavelengths of light are absorbed by specific tissue components.
Together, absorption and scattering determine the amount and spectral distribution of the light that is reflected. Hence, diffuse optical images do not only give information on the shape and the surface of the material imaged, but more importantly on what is below the surface.
Depending on the wavelength of the light, sampling depths in the order of, or up to, a centimeter, cm, are feasible.
In multispectral or hyperspectral imaging, optical images are acquired using dedicated camera systems that detect light in different wavelength bands. The images taken by these cameras are often processed by various image processing methods to obtain information on the status of biochemical composition of the imaged object. Examples of materials subjected to diffuse optical imaging range from raw foods to assess their freshness, crops to assess their status, to live human tissues to detect and locate remaining cancer cells during and after surgery.
Although light scattering is the very phenomenon that enables diffuse optical imaging, it also complicates quantitatively accurate imaging, and it limits the sharpness of the images taken.
When imaging a homogenous medium, the diffuse light captured in the form of diffuse reflection gives an excellent representation of the optical properties of the tissue. If images are taken at sufficient wavelengths, then there are ample mathematical models available that allow spectral unmixing, i.e. the extraction of the different concentrations of the tissue components.
These methods vary from complicated diffusion theory based spectral analyses based on a lot of prior knowledge on light transport and the tissue components to purely mathematical techniques such as spectral unmixing, SU, or machine learning, ML. Often, more simple approaches are used such as ratioing the diffuse reflectance at two strategically chosen wavelengths.
Preferentially, however, diffuse optical imaging is used in cases where a defect is not spread over the tissue evenly, and its location is not known. In fact, the purpose of imaging usually is to detect and locate a defect. In such a case the diffuse nature of the reflected light limits the sharpness of the image, and thereby decreases the accuracy of detection of small defects.
In diffuse media, the sampling depth and imaging resolution are inversely related: in the case of a decreased scattering coefficient the diffuse light detected has sampled deeper into the tissue, generating an image with a lower spatial resolution, while a sample with a larger scattering coefficient samples more superficially and may have a higher spatial resolution.
A case of inhomogeneously distributed optical properties is not rare, since most biological materials are very inhomogeneous. A possible defect can be considered as an additional inhomogeneity. For diffuse light this has serious consequences as the diffuse light will encounter different optical properties at different locations. Because these optical properties are wavelength dependent and this wavelength dependence is different for different tissue compositions, the diffuse light will have different distributions at different wavelengths.
As a consequence, different wavelengths will sample different volumes. This will compromise any spectral unmixing approach. This phenomenon is very well known in the field of diffuse optical spectroscopy.
In many cases, simple ratios of images taken at strategically chosen wavelengths can give excellent information on the ratio between different absorbing components. These numbers can then be used to characterize the tissue.
As an example, a fat-to-water ratio image has been proposed to be of particular interest, in particular for a diagnosis of breast cancer. Such a ratio would be accurate in homogenous tissue. Due to the inhomogeneous nature of the distributions of these tissue components, the ratios calculated have been documented to be in error by as much as 50%.
For a medium, the diffuse reflection Rd can be related to the absorption coefficient by the relation:Rd˜  (1.1)where μa stands for the absorption coefficient and l for the average path length of the detected photons.
When taking two wavelengths λ1 and λ2, a ratio X can be calculated from the natural logs of the diffuse reflections:
                    X        =                                                            〈                                  l                  2                                〉                            ⁢                                                μ                  a                                ⁡                                  (                                      λ                    2                                    )                                                                                    〈                                  l                  1                                〉                            ⁢                                                μ                  a                                ⁡                                  (                                      λ                    1                                    )                                                              =                                    ln              ⁡                              (                                                      R                    d                                    ⁡                                      (                                          λ                      2                                        )                                                  )                                                    ln              ⁡                              (                                                      R                    d                                    ⁡                                      (                                          λ                      1                                        )                                                  )                                                                        (        1.2        )            
Assuming the presence of two absorbers at molar concentrations C1 and C2, then:μa(λ)=C1μa,1(λ)+C2μa,2(λ)  (1.3)where μa,m(λ) stands for the molar absorption coefficient of absorber with index m. Introducing the concentration ratio:
                    ψ        =                              C            1                                              C              1                        +                          C              2                                                          (        1.4        )            so that (1.3) can be written as:μa(λ)=(C1+C2)(ψμa,1(λ)+(1−ψ)μa,2(λ))  (1.5)
Now expression (1.2) can be rewritten into:
                    X        =                                            ln              ⁡                              (                                                      R                    d                                    ⁡                                      (                                          λ                      2                                        )                                                  )                                                    ln              ⁡                              (                                                      R                    d                                    ⁡                                      (                                          λ                      1                                        )                                                  )                                              =                                                    〈                                  l                  1                                〉                                            〈                                  l                  2                                〉                                      ⁢                          (                                                                    ψ                    ⁢                                                                                  ⁢                                                                  μ                                                  a                          ,                          1                                                                    ⁡                                              (                                                  λ                          2                                                )                                                                              +                                                            (                                              1                        -                        ψ                                            )                                        ⁢                                                                  μ                                                  a                          ,                          2                                                                    ⁡                                              (                                                  λ                          2                                                )                                                                                                                                  ψ                    ⁢                                                                                  ⁢                                                                  μ                                                  a                          ,                          1                                                                    ⁡                                              (                                                  λ                          1                                                )                                                                              +                                                            (                                              1                        -                        ψ                                            )                                        ⁢                                                                  μ                                                  a                          ,                          2                                                                    ⁡                                              (                                                  λ                          1                                                )                                                                                                        )                                                          (        1.6        )            
From expression (1.6) it is clear that, when the path lengths l1 and l2 are identical and the absorption properties of both absorbers are known, the concentration ratio ψ can be derived from the ratio of the natural logarithms of the reflectances. The strongest sensitivity is found when wavelengths λ1 and λ2 are chosen to be at wavelengths where the two absorbers have maximum differences. This ratio is often used in hyper or multi spectral imaging to quantify concentration ratios of chromophores.
The main problem in practical application of expression (1.6), lies in the assumption that l1=l2. In practice the path lengths, l1 and l2, are dependent on the optical properties.
There are three main reasons why the assumption l1=l2 is problematic in practical use.
A first reason is that the path length is dependent on both the absorption and the scattering properties. When aiming at a maximum sensitivity, the wavelengths λ1 and λ2 are chosen to be at wavelengths where the two absorbers have maximum differences. This results in very different path lengths and sampling volumes, even in homogeneous tissue.
A second reason is that the two tissue types with different concentrations of the two absorbers considered may be biologically very different. Hence it is likely that these tissues will also have very different scattering properties. This too causes large differences in path lengths and hence differences in sampling volumes.
A third reason is that the need for imaging implies inhomogeneous tissue, where different types of tissue with very different optical properties coexist. This results in very different path lengths and very different sampling volumes in the different tissue areas. Also it can generate artefacts in the boundary region between the two tissues. Detected photons show a preference for ‘paths of lowest absorption’: Light detected at the wavelength of maximum absorption of a first absorber will have a stronger contribution from photons that have travelled preferentially through tissue containing a higher concentration of a second absorber, and vice versa. So, even if l1 would be of the same length as l2, the two paths would have sampled very different tissue volumes. This problem leads to an overestimation of the total amount of tissue components.