The ability to measure the mechanical properties of highly viscoelastic materials such as cells and soft tissues requires observation of the instantaneous (elastic properties) and time dependent (viscous properties) properties of tissues and materials at low strains where the material will be in the linear region of the stress-strain curve. For soft polymeric materials like silicone rubber this is not a requirement since the viscous component of the modulus is approximately constant and contributes only about 10% to the elastic modulus values measured [14]. However, this is not true for soft tissues since when they change their shapes and exude fluid into the interstitial spaces at low strain rates which causes a change in Poisson’s ratio. This behavior influences mechanotransduction pathways leading to changes in cell division, changes in DNA, and protein synthesis [16]. At low strain rates, viscous behavior can alter the modulus value up to about 50% due to fluid movement and cellular behavior during mechanical loading [16]. One way of quantitatively assessing these changes is by measuring the changes in the resonant frequency of a material at different strain rates [19][23]. The resonant frequency of tissue is the frequency at which the material undergoes its maximum displacement and energy storage occurs in almost a purely elastic manner.

Resonance is a phenomenon in physics, that defines that a system will oscillate with a greater amplitude at specific frequencies [27]. These frequencies, known as resonant frequencies, represent inherent properties of the material determined by the physical characteristics and configuration [2]. At the resonant frequency of a polymeric material large amounts of energy are stored in reversible backbone bond angle and molecular length changes [28]. In collagenous tissues this occurs by stretching the flexible regions at the molecular and fibrillar levels [29] Measurement of tissue component resonant frequencies invivo, including that of cells and other macromolecular structures, provides “real life” values of the elastic modulus which are different from the mechanical properties of isolated cells and macromolecules.

The maximum displacement of a mechanically loaded sample occurs during elastic deformation. This condition is where little energy is lost due to friction, fluid flow, or molecular sliding. Using optical coherence tomography (OCT) changes in tissue dimensions can be obtained instantaneously at varying applied deformations [17][18]. Using OCT an image is formed from reflected infrared light from the sample. At deformation rates up to as high as several thousand cycles per second instantaneous deformation can be measured from the image formed [17][18]. From studies at different deformation rates the resonant frequency can be determined from the frequency at which the maximum deformation occurs. In multilayered materials each layer will have its own resonant frequency characteristic of the properties of that part of the tissue [19][20]. In skin the epidermis, papillary collagen, and blood vessels all have characteristic resonant frequencies that change because of skin pathology [19][20].

Measurement of the resonant frequency of a tissue can then be related to the composition by reviewing the images created by OCT at different loading frequencies [19][20]. Calibration of VOCT measurements on a material is accomplished by: (1) instantaneous stress-strain measurements of the behavior of a tensile sample measured using a load cell and micrometer; and (2) by measuring the resonant frequency and thickness of the same sample at the same time [19][20]. These measurements can then be used to create a calibration curve that leads to the development of a relationship between E, the elastic modulus, the sample thickness d, and the resonant frequency, fn, as shown in equation (1) for soft tissues and synthetic polymers with densities close to 1.0.

The relationship is developed by comparing tensile stress-strain measurements (E) to resonant frequency (fn) measurements on the same material at the same time for sample thickness (d) measured from the OCT image or with a micrometer (see equation 1) [19][20]. VOCT measurements made on synthetic polymers and soft tissues (see Table 1) were used to develop equation (1):

SoftTissues: Exd = 0.0651 × ( f n 2 ) + 233.16 (1)

Figure 1. (A) Block diagram of the setup of an OCT device modified to do vibrational optical coherence tomography (VOCT). The speaker shown is about 2 inches in diameter and provides audible sound through a computer-driven app to vibrate the sample between 30 and 300 Hz for skin. The displacement of the sample at each frequency is obtained from amplitude data collected and stored from raw images created by the OCT. The sample displacement is divided by the displacement induced by the speaker in the absence of the sample to correct for speaker induced vibrations. (B) Schematic drawing of the OCT handpiece used to collect the VOCT data at each frequency on skin and other soft tissues invivo.

Resonant frequency measurements and elastic modulus calculations on different tissues can be collected by vibrating the sample using sound from an acoustic speaker at fixed frequencies between 30 and up to 1000 Hz and then collecting the instantaneous image of the sample at each frequency [19][20]. The measurement reported in this paper are synthesized into final form from previous publications [13]-[26]. During resonant frequency measurements the sample is continuously loaded by an acoustic signal generated by a speaker through an app on the OCT unit. Figure 1 shows a block diagram of the system used to measure the resonant frequency of a tissue. Typically for soft tissues acoustic vibrations between 30 Hz to 500 Hz are used to measure the resonant frequency (s) of most soft tissues. While the applied vibrational frequency and sample vibrational frequency are the same, the resulting sample component deformation will depend on the sample material properties. The displacement of the sample at these frequencies is very small. Collagenous soft tissues operate in vivo in the linear low modulus region of the stress-strain curve [21][22]. For collagenous tissue such as skin and cornea, the in vivo strain has been estimated to be about 5% [21]. A strain between 5% and 10% falls into the low modulus or toe region of the collagen tensile stress-strain curves of these tissues [21]. At strain rates above 100 Hz (the resonant frequency of collagen in many soft tissues in vivo), the viscous component decreases and approaches a value of about 5% of the value of elastic modulus at high frequencies. At high frequencies the modulus becomes almost purely elastic as has been shown for skin and cornea [15][17]. The modulus calculated from the resonant frequency is due to storage of elastic energy which reflects only bond stretching and not molecular and fibrillar slippage as well as rearrangement of the attached and surrounding fluid which is responsible for the viscous component of the behavior [15][17]. At low frequencies intermolecular and interfibrillar fluid flow and cellular deformation behavior in living tissues is associated with a large viscous component. At high frequencies above 100 Hz the fluid and cells do not have enough time to alter their shapes and location and the behavior becomes almost purely elastic.

Figure 2. Weighted displacement versus frequency for decellularized human dermis. The frequency at which the maximum displacement of a sample occurs is defined as the resonant frequency and is related to the elastic modulus of a material (see equation 1). Note the sharpness of the peak defines the material uniformity. The weighted displacement versus frequency data shown is for decellularized human dermis, which has a modulus of about 2 MPa at a strain of 5% invitro. The weighted displacement is the displacement of the sample divided by the displacement in the absence of the sample due to speaker vibrations. Note that the elastic modulus is estimated by using equation (1) after the thickness (d) is measured either using a micrometer or from OCT images. Measurements on purified dermal collagen, skin, cartilage, cornea, silicone rubber and other synthetic polymers were used to develop equation (1). VOCT measurements and tensile measurements made on the same material at the same time were used to verify the relationship.

The easiest way to explain how resonant frequency data is measured is to start with a simple system that is composed of pure collagen. Decellularized human skin is made up of collagen types I and III after the cells and DNA are removed [21]. Using this material the resonant frequency can be measured by vibrating the sample with sound from an acoustic speaker at frequencies between 50 and 300 Hz. The resonant frequency is the frequency at which the maximum displacement is observed which in this case is about 150 Hz (see Figure 2). The weighted displacement is the ratio of the observed displacement with the sample present divided by the displacement without the sample present. The speaker has its own resonant frequencies that must be considered when testing a tissue sample. Once the resonant frequency is measured the elastic modulus can be calculated using equation 1 after the sample thickness is measured.

The same approach can be used to measure the resonant frequency of skin invivo which is composed of cells, papillary collagen, and blood vessels. Figure 3 shows the weighted displacement versus frequency for normal skin and scar tissue illustrating the increased resonant frequency seen in scar tissue compared to normal skin [22]. Note cells and blood vessels in normal skin invivo are usually minor components of the tissue and do not show displacement peaks at 50 and 150 Hz in some locations. Pathological scar tissues are characterized by altered cross linking making them stiffer [22]. This explains the increased resonant frequency difference seen when comparing normal skin and scar.

Figure 3. Weighted displacement versus frequently for human skin (left) and scar (right) measured invivo. The elastic modulus of normal skin with a resonant frequency of 90 to 100 Hz is about 2.5 MPa and that of scar with a resonant frequency of about 240 - 250 Hz is between 6 and 10 MPa. The resonant frequency of normal skin is much lower than that of neighboring scar tissue. Using this approach the resonant frequency of a variety of materials, cells, and tissues can been characterized invivo.

When the elastic modulus is studied for a variety of tissues using VOCT it has been observed that each tissue component has a “characteristic” modulus that is observed for each tissue component (see Figure 4 and Table 1). The cellular resonant frequency of normal epithelial and most cells invivo is about 50 - 60 Hz yielding an elastic modulus of about 1.0 MPa in skin and cornea [20][22]. Cancer associated fibroblasts are somewhat stiffer in basal and squamous cell carcinomas, and melanoma. These fibroblasts have a resonant frequency invivo of about 70-80 Hz and have moduli of about 1.2 MPa [23]-[25]. This contributes to the stiffening of cancerous skin lesions. There is a proliferation of normal epithelial cells and cancer associated fibroblasts in all these three types of skin cancers [23]-[25]. Cancer associated fibroblasts are stiffer than normal cells because of increased connections with neighboring cells and fibrotic tissues compared to normal epithelial cells.

Table 1. Resonant frequency peaks and their associated moduli based on VOCT measurements [26].

Note: SD = standard deviation.

The viscous behavior of cells is much greater than the viscous behavior of collagen in tissues and contributes to energy dissipation that prevents soft tissue injury and failure. Loss of cells and proteoglycans from decellularized human dermis transform skin from a flexible material into a brittle stiff material that fails at lower strains. This partially explains the loss of skin’s ability to resist tearing in older individuals where collagen fiber fragmentation and loss of proteoglycans occur.

Synthetic polymers and natural tissues all have characteristic resonant frequencies that differ based on the differences in the molecular and supramolecular compositions of the materials [20]. Since each tissue has a “characteristic” resonant frequency and elastic modulus, the mechanical properties of cells and tissues in different disease states can be measured and related to changes in cellular and tissue compositions.

Figure 4. Plot of log of the elastic modulus for cells, tissues, and synthetic polymers derived from VOCT measurements made both invitro and invivo. The relationship between E and d for the different materials were determined experimentally and only soft tissues follow equation (1) since the density of bone and synthetic polymers is different than 1.0 gm/cc.

One of the important uses of VOCT is to follow the changes in mechanical properties of tissues associated with progression of diseases noninvasively. A good example of this is the stiffening of both cancer associated fibroblasts and fibrotic tissue associated with skin cancer formation as well as softening of new thin blood vessels. VOCT measurements made on normal skin invivo are characterized by several peaks (see Figure 5). In normal skin, a small cellular peak occurs usually at about 50 Hz, a papillary collagen peak at about 100 Hz, a blood vessel peak at about 150 Hz. Sometimes fibrous peaks are seen at 180 - 250 Hz in skin cancers [23]-[25] (see Figure 5 and Figure 6).

Figure 5.(A) A 2D color-coded OCT image of a cross-section of normal skin invivo characterized by a yellow stratum corneum layer, pink and red granulating cell layers, and blue papillary and basal cell layers. (B) A plot of weighted displacement versus frequency for normal skin. Note the peaks at about 50 Hz (cells), 100 Hz (dermal collagen), and 150 Hz (blood vessels) are found in normal skin. The peak at 220Hz reflects some fibrous tissue present. The cells are found primarily in the epidermis and papillary dermis, while the dermal collagen and blood vessels are found in the papillary dermis based on tissue histology [25].

Figure 6. OCT image and VOCT weighted displacement versus frequency plots from a lesion diagnosed as a melanoma by a board-certified dermatopathologist. (A) 2D color coded cross-sectional OCT image of a cancerous melanoma (see circle). (B) Plot of weighted displacement versus frequency showing the mechanovibrational peaks. The change in size of the peaks at 80 Hz (cancer associated fibroblasts), 130 Hz (new thin blood vessels), and fibrous tissue at about 260 Hz are characteristic of all types of skin cancers including melanoma, basal cell carcinomas, and squamous cell carcinomas. The color coding of the OCT image was accomplished based on the pixel intensity and use of a look-table to assign the colors. The diagnosis of this lesion was based on histopathological review of the lesion by a board-certified dermatopathologist. Note the mechanovibrational data in B used to characterize cancerous lesions is supported by convolutional neural networks and AI analysis used to characterize the cancers [25].

When VOCT is used to study cancerous tissue the color-coded OCT image (see Figure 6(A)) and spectrum of resonant frequencies change (compare Figures 5(B) andFigure 6(B)). The color coding of the OCT image was accomplished based on the pixel intensity and a look-table to assign the colors. There are new peaks at 70 - 80 Hz (cancer associated fibroblasts), new thin cancer associated blood vessels (130 Hz), and a fibrotic collagen peak occurs at 250 - 260. The change in resonant frequency of cells (80 Hz), new thin blood vessels (130 Hz), and fibrotic collagen (250 - 260 Hz) is observed in all cancerous lesions including BCC, SCC, and melanoma.

Figure 7. A plot of the loss modulus determined from VOCT relaxation experiments as a percentage of the elastic modulus of decellularized human dermis invitro as a function of vibrational frequency and strain. Note the loss modulus of decelluarized dermis does not change much with increasing strain. The loss modulus is low for pure collagen fibers in decellularized human skin compared to normal skin invivo and is like that of synthetic rubbery polymers. This viscous behavior of stretched collagen fibers is like that seen for silicone rubber (data not shown); the loss modulus of silicone rubber is almost independent of strain and is about 10%. Note the absence of proteoglycans and cells in decellularized dermal collagen reduces the viscous component and the viscoelastic behavior (see Figure 8). The data shown was collected on 5 samples of decellularized dermis.

To measure the viscoelasticity of soft tissues and polymers the VOCT loading conditions are different than for elastic tissue measurements. In the viscoelastic experiment for determination of the viscous response, the sample is cycled through at least three acoustic loading cycles and then the acoustic loading is terminated. The loss modulus is estimated from the half width of the decay of the weighted displacement versus time. The viscous component is obtained by dividing the change in frequency at the half height of the mechanovibrational peak, or 3 decibels down from the maximum peak in the power spectrum, by the driving frequency. This method is known as the half-height bandwidth method [22]. This approach follows the delay in the relaxation which is related to the viscous behavior of the sample. A longer decay in a viscoelastic material is associated with a larger half width of the weighted displacement versus frequency curve. Figure 7 shows the relationship between loss modulus and frequency for decellularized human skin in vitro at different applied strains.

Figure 8 shows the loss modulus for human skin measured noninvasively invivo using VOCT [22]. The loss modulus is greater in normal skin invivo compared to decellularized human dermis invitro. The increased viscous component of normal skin compared to decellularized dermal collagen is likely due to cell contractile forces exerted on cells connected in series together and due to the presence of proteoglycans. In addition, the influence of proteoglycans present of the surface of collagen fibrils and fibers provides some interfibrillar slippage. The rearrangement of intermolecular and interfibrillar fluid also contributes to the viscous behavior at low strain rates. This increased viscous behavior at low strain rates prevents skin from tearing during stretching and dissipates applied energy.

Figure 8. Loss modulus of normal skin measured using a VOCT relaxation test conducted invivo. The data for the loss modulus is shown as a fraction of the elastic modulus versus frequency in Hz for normal human skin. Note the maximum loss is found for skin components at frequencies less than 50 Hz. The high loss modulus observed at low frequencies is a result of rearrangement of the viscous interfibrillar materials, fluid flow, and the cellular contribution to the viscoelasticity of skin. At low strain rates cells can respond to provide a response to an applied load whereas at high strain rates the cells behave as rigid elements and cannot dissipate enough energy to prevent skin tearing. Viscous energy loss by muscle and tendon invivo is like that of skin and is an important means of energy dissipation in soft tissues that limits premature mechanical failure. This also is true in the cornea which protects the components of the anterior and posterior segments of the eye from low frequency mechanical damage [15] (see Figure 9).

Figure 9. A plot of loss modulus as a fraction of the elastic modulus versus frequency for whole pig globe from measurements on the cornea versus excised cornea invitro determined using VOCT. These measurements were conducted on at least 5 samples of each tissue after pulsing the sample with acoustic sound and then making measurements directly after removing the acoustic signal. Note the loss moduli decrease with increasing frequency and is similar in pig globes and excised corneas and in human tissue invivo. At low mechanical loading frequencies, the cornea is highly viscoelastic and dissipates forces and energy applied to the eye. Therefore, mechanical measurements made at frequencies below 100 Hz need to be corrected for frequency and strain-rate dependence.