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39CHARACTERIZATION OF TISSUE ELASTICITY WITH OPTICAL COHERENCE ELASTOGRAPHY: GOING BEYOND THE LINEAR PARADIGM
VLADIMIR Y. ZAITSEV1, ALEXANDER A. SOVETSKY1, LEV A. MATVEEV1, ALEXANDER L. MATVEYEV1, EKATERINA V. GUBARKOVA2, ANTON A. PLEKHANOV2, MARINA A. SIROTKINA2 AND
NATALIA D. GLADKOVA2
1Institute of Applied Physics of the Russian Academy of Sciences, Russia 2Institute of Experimental Oncology and Biomedical Technologies, Privolzhsky Research Medical University, Russia
vvuzai@ipfran.ru
Abstract
In this report we present recent results related to characterization of nonlinear elastic properties of biological tissues using quantitative Compression Optical Coherence Elastoraphy (C-OCE) with application of reference silicone layers as optical stress sensors. In the reduced form this approach has already proven its efficiently for estimation of the tangent Young's modulus of nonlinear tissues for a standardized pressure. Analysis of entire stress-strain dependences opens new possibilities for diagnostics of biological tissues, which was inaccessible within the framework of conventionally used paradigm of linear elasticity.
C-OCE is an emerging approach to studying mechanical properties of biological tissues [1]. The idea of this approach was borrowed from the medical ultrasound elastography [2] and was introduced in OCT by J. Schmitt [3] based on the paradigm of linear elasticity of the tissue. Within the framework of this paradigm it is supposed that the values ofuniaxial stress c the values of stress and the resultant axial strain s are linearly proportional to each other:
c = Es (1)
The axial strain in OCT can be found by comparing subsequently acquired OCT scans, namely, by estimating axial gradients of interframe phase variations using either classical least-square procedures [4] or recently proposed more advanced "vector" method [5, 6].
It is clear from Eq. (1) that in order to quantify modulus E , knowing only local strain eis not sufficient, so that estimation of stress c is also required. In principle, one can try to evaluate stress c by measuring the force applied to the tissue from the OCT probe with a known contacting area. However, in [2] and [3] it was proposed that thespatialdistribution of the Young's modulus (at least in the relative sense) can be extracted from the initially reconstructed axial profiles of strain. Indeed, if the deformation is produced by uniaxial stress (i.e., the tissue is allowed to freely expand laterally) the stress c is the same for different depths with the Young's moduli E12 and
the corresponding strains s12 of the tissue, so that
c = E1s1 = E2S 2, (2)
or equivalently
E1/ E2 =S 2/ S1 (3)
The measurement of the axial strain distribution, therefore, is the key step in deconstructing the distribution of the Young's modulus. Equations (2) and (3) mean that the depth distribution of local strains is directly proportional to relative distribution of inverse Young's modulus 1/ E . Therefore, if in some area this modulus is known, this opens the possibility to quantify the elastic modulus over the entire imaged region. This idea was mentioned quite long ago [2], but did not found application in ultrasound compression elastography, whereas in compression OCE the utilization of translucent reference layers (usually made of soft silicone) is efficiently used for quantifying the tissue elasticity [7,8]. Although in C-OCE typical values of interframe strain are on the order of 10D4..10D3, for which the linear Eqs.(2)and(3) hold fairly well, quite often larger strains (up to several per cent and greater) are produced in the tissue. In many cases the tissue surface is not ideally plane, so that even if a highly compliant silicone layer is placed between the OCT-probe window and the sample, the mechanical contact over the entire visualized zone is attained only when the most prominent tissue areas are strained up to several per cent. With further compression the strain becomes even greater. Such strains cannot be directly measured by comparing individual OCT scans, but can be estimated by analyzing a series of scans acquired during the tissue deformation and finding cumulative strains for such a series [8, 9]. For such larger strains, biological tissues exhibit rather pronounced nonlinear elasticity [10]. In such a case the stress-strain dependence c(s)for the tissue is essentially
nonlinear, whereas linearized Eq.(1) may hold only for sufficiently small increments in stress Act , and strain As , so that the elastic modulus becomes dependent on the tissue pre-compression,
E(ct) = Act/ As (4)
To measure nonlinear stress-strain dependences ct(s) , one needs to measure simultaneously strain in the tissue and the applied stress. In view of the above-mentioned fact that quite often the stress in the visualized region depends on the lateral coordinate, a method is needed to estimate the current local stress during the tissue deformation. In this regard an important question arises whether it is possible to use reference silicone layers as stress sensors not only for very small strains, but for larger strains as well. In other words, how strong is the nonlinearity of silicone? To verify this point the straightforward solution is to independently measure stress and strain, which is quite challenging because of numerous possible distorting factors (such as the influence of stiction between the OCT-probe and silicone as discussed in [8]. However, there is another solution proposed in [3]. It is based on utilization of a sandwich structure made of two siliocones with highly contrasting strains. During the compression of this sandwich one may compare the cumulative strains in two silicones by plotting one strain against the other to verify is this dependence is linear or nonlinear. The stiffer silicone is deformed weaker (still remaining in the definitely linear region of straining), whereas the softer experiences significantly stronger deformation and thus can potentially exhibit pronounced nonlinearity, whereas for linearly deformed materials, the strains should remain proportional to each other resulting in a linear dependence. It has been experimentally verified that strains of two silicones in such compressed sandwich structures demonstrate very good linear proportionality up to quite large strains over 50% (see Fig. 1a).
Figure 1: (a) Illustrain of the self-testing of silicones' linearity using compression of a sandwich composed of two silicone layers with contrasting stiffness. (b) An examples of nonlinear stress-strain curve for normal breast stroma obtained using a pre-calibrated silicone layer as the stress sensor.
This experimental finding indicates that silicones are rather linear materials, so that even for such fairly large strain in the reference silicone layer, its strain remains linearly proportional to stress. This means that pre-calibrated reference silicone layers may serve as optical stress sensors and can be used to local stress over the visualized region. Thus measuring cumulative strains in the reference silicone layer and plotting it against the cumulative strain in the tissue one can obtain nonlinear strass-strain curves for the examined tissue sample. An experimentally obtained example of such a curve is given in Fig. 1b corresponding to a normal stroma of human breast tissue. This figure demonstrates that for strains below ~15% the stress-strain dependence is rather linear with almost invariable slope (i.e., initially the Young's modulus is nearly invariable ~30kPa), whereas for larger strains the slope of ct(s) noticeably increases, so that theodulus becomes several times greater tending to ~150 kPa for strains >35%). It can be pointed out that for other components of breast tissue (pre-cancer states like fibrosis and hyalinosis and, furthermore, for cancerous tissues) similar stress-produced stiffening can be observed for several times smaller strains (even for strains ~1.5-2%) [8].
In view of such pronounced nonlinearity, unambiguous quantitative interpretation of C-OCE-based estimates of the elastic moduli and meaningful comparison with other measurements requires that one should specify the pressure, for which a particular value of the Young modulus is obtained. Even if a sample is strongly mechanically inhomogeneous and stress during the C-OCE examination is also inhomogeneous, it is possible to construct an OCE-image for a selected standardized pressure by processing a series of OCT images obtained during the sample compression (see details in [11]). Examples of quantitative interpretation of C-OCE data for the tissue stiffness estimated using the pressure standardization can be found in refs. [12,13,14].
In studies [12,13,14], however, the ability of the proposed method to obtain nonlinear stress-strain curves was utilized in the reduced form, only for estimation of the tangent (current) Young's modulus for a chosen standardized pressure. However, in fact the analysis of entire nonlinear stress-strain dependences suggests new diagnostic information. In particular, nonlinear curves can be fitted using one or another analytical law, such that in addition to
the Young's modulus, parameter(s) characterizing the tissue nonlinearity can be also extracted and used as additional diagnostic signatures of the tissue state (including classification of pathologies). To demonstrate diversity of nonlinear elastic properties for various types of biological tissues Fig. 2 shows examples of stress-strain curves and dependence of the tangent Young's modulus as a function of tissue strain for three significantly different samples: an excised fragment of human artery wall, excised rabbit's cornea and rat's cortex (in normal state). For cornea and the artery wall, the nonlinearity looks as pronounced tissue stiffening with increasing compression strain. In contrast, for cortex, the nonlinearity demonstrates qualitatively different character: the opposite sign corresponding to the decrease in the Young's modulus with increased compressive strain. In that experiment it was verified that the white matter and the tumor (astrocytoma) region also demonstrated qualitatively similar initial softening with increased compression, which was changed to slight stiffening with further compression. It is interesting to point out that in contrast to majority of other tumors (for which the Young's modulus is usually much higher than for the normal tissue), astrocytoma demonstrated initially more than twice lower stiffness than normal cortex and white matter. However, after fairly moderate ~5% strain astrocytoma demonstrated pronounced (about two times) stiffening, such that its Young's modulus could become comparable with its values for normal cortex and white matter. These results indicate that analysis of full nonlinear stress-strain dependences gives much richer information for differentiation of tissue types in comparison with conventionally discussed linear elasticmodulus.
(a) (b) (c)
Figure 2: (a) examples of structural OCT scans for three different tissues (excised rabbit cornea, fragment of human artery wall and rat's cortex). (b) are the corresponding nonlinear stress-strain curves obtained by the C-OCE method using reference silicone layer as a linear stress sensor. (c) are the dependences on tangent Young's modulus derived from the stress-strain
curves shown in panels (b).
Although the behavior of cortex (as well as white matter and astrocytoma) with atypical softening differs from the nonlinear behavior of most tissues, it can be pointed out that such nonlinearity is known for materials containing microstructural heterogeneities with threshold reaction to loading, for example, it was observed for an artificial material in the form of a homogeneous gel-like matrix with embedded hollow shells that experience buckling when compressive loading reaches certain threshold value [15]. Probably microstructural units with similar mechanical properties may occur in some naturaltissues.
The nonlinearity of stiffening type (like for cornea and the vessel wall) is easier to understand. Such tissues with collagenous matrix are penetrated by narrow pores/gaps (often crack-like ones) through which interstitial physiological liquids penetrate, in particular to supply the tissue with nutrition. Such gaps are more compliant than the surrounding matrix tissue and with increasing compression these gaps gradually close, so that their concentration gradually decreases and the tissue becomes more dense and stiff. Such behavior is clearly demonstrated by cornea, in which the gaps among collagenous layers become nearly completely closed by reaching average strain ~5-7%. For such compression, the stiffness of cornea increases by an order of magnitude and becomes comparable with stiffness
of cartilaginous tissue. The compliance properties of such narrow gaps/pores resemble the properties of cracks that attract much attention in nondestructive testing, geophysics and related areas, where it is shown that the average strain producing closing of such narrow gaps is determined by their aspect ratio (i.e., the ration of the opening to the lateral size). The models describing crack-containing materials can be adapted to describe softening/stiffening of biological tissues due to the influence of interstitial gaps, such that the C-OCE data can be used to estimate averaged aspect ratio of such gaps [16] and even to reconstruct the distribution of these gaps over their aspect ratios [17].
Although the fact of pronounced elastic nonlinearity of biological tissues has been known over several decades, in OCE the linear-elasticity paradigm still remains dominating and the OCE-based methods for obtaining nonlinear stress-strain curves and their utilization for diagnostic applications are yet emerging (e.g., another variant of OCE-based method for obtaining stress-strain curves can be found in [18]). However, bearing in mind the recent breakthrough results on the development of C-OCE techniques [1] one can expect a similar breakthrough in the development of C-OCE beyond the linear paradigm, which should significantly extend diagnostic potential and accuracy of OCT-based elastographic methods.
This work was supported by RSN grant 16-15-10274 in part of the development of C-OCE method and RFBR grant 18-32-20056 in part of the analysis of nonlinear stress-strain dependences.
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