In order to assess the validity of MM-OCE for measuring dynamic elastic properties of materials with different stiffness, we examined a set of polymer-based samples with elastic mechanical properties that spanned a wide range. Polydimethylsiloxane(PDMS)-based samples with optical and mechanical properties representative of soft polymers, composites, and biological tissues were prepared [4
]. To vary the elasticity of the samples, the ratio (PDMS):(curing agent RTV 615 A) was varied in the range 25:1–10:1, while the relative ratio of the curing agent and the crosslinker, RTV 615 A:RTV 615 B, was maintained at 10:1 for all samples. Titanium dioxide (TiO2
) microparticles (Sigma-Aldrich, #224227, average diameter 1 μm, < 5 μm) with a concentration of 4 mg/g served as optical scatterers and magnetite MNPs (average diameter 25 nm, Sigma-Aldrich, #637106) with a concentration of 2 mg/g served as magnetomotive perturbative agents. The polymers and the TiO2
particles were thoroughly mixed with the aid of a magnetic stir plate. Subsequently, the MNPs were added and the sample solution was homogenized in an ultrasonic sonicator for five hours. They were then poured into Petri dishes with a diameter of 38 mm and a height of 5 mm, curing 12 hrs at 80°C, and subsequently 24 hr at room temperature (22 °C). An indentation instrument (model TA.XT Plus Texture Analyzer, Texture Technologies Corp., Algonquin, IL) was used for independent validation measurements of the elastic moduli of the samples.
The composition of these samples was varied, resulting in noticeable (by palpation) differences in their elasticity. As confirmed by indentation measurements using a commercial indentation instrument, the static elastic moduli of these samples ranged from 0.4–140 kPa. The concentration of optical scatterers and MNPs as well as the geometry and dimensions of all samples were kept constant. In this study, the MNPs are believed to be mechanically bound directly to the solid polymer matrix of the silicone medium. The consistency and repeatability of the MM-OCE measurements support this hypothesis.
A spectral-domain optical coherence tomography (OCT) system [23
] was used to perform real-time interferometric imaging of the samples. The sample arm was modified to accommodate a small electromagnetic coil custom-designed to optimize the magnetic field gradient within the focal region of the optical imaging system, as illustrated in . The probing light was provided by a Nd:YVO4
-pumped titanium:sapphire laser (KMLabs, Boulder, CO) with a center wavelength of 800 nm and a bandwidth of 120 nm, providing an axial resolution of 3 μm in the samples. The average power incident on the samples was 10 mW. A 40 mm focal length lens was used to focus the light in the sample arm to a 16 μm spot (transverse resolution). Magnetic field modulation was synchronized with optical data acquisition. M-mode imaging data was acquired at a camera line rate of 29 kHz (34 μs per axial depth scan) for a total acquisition time of 280 ms per M-mode image. During the acquisition of each image the magnetic field was turned on shortly after the start of the acquisition and kept constant for 100 ms, and then switched off in a square-wave pattern, releasing the MNPs and resulting in the relaxation of the sample, as illustrated in .
Fig. 1 Schematic diagram of the MM-OCE set up with MNPs. (top left) The magnetic coil provides a magnetic field that is aligned axially with the imaging beam. The field gradient engages the motion of MNPs in the sample. (top right) Transmission electron micrograph (more ...)
Fig. 2 Scatterer response upon square-wave modulation of a magnetic field. (a) Two-dimensional (x-z) cross-sectional (B-mode) amplitude OCT image of a silicone sample containing MNPs and TiO2 optical scatterers. The dashed line indicates the location in the (more ...)
The complex analytic signal obtained from the raw optical data acquired in M-mode was used to extract the phase associated with individual scatterers in the samples (at fixed positions in depth) as a function of time. The phase variation was recorded as the magnetic field was applied step-wise to the sample, and the absolute displacement of the scatterers was deduced from Eq. (1)
is the time between consecutive axial scans, dϕ(dt)
is the change in unwrapped phase (in radians [29
is the index of refraction of the silicone sample medium (1.44, as measured by OCT [30
is the center wavelength of the probing light (800 nm), and dz(dt)
is the displacement of each scatterer over dt
. Based on the parameters of our system and of our samples, the displacement can be calculated directly from the equation above. The displacement sensitivity of the system, defined as the standard deviation of the measured position of a stationary mirror in the sample arm, was 11 nm. Typical maximum displacements measured in the samples were in the order of a few hundred nanometers.
Within 2–20 ms after the onset of the magnetic field, the scatterers were observed to reach a maximum displacement. This was followed by an underdamped oscillation (). If a steady magnetic field gradient was present, such as following a step-function change in the applied magnetic field gradient, the scatterers eventually settled to a new static position as they reached a new equilibrium position (data not shown). A similar behavior was observed when the field was removed and the MNPs in the sample were released from the magnetic force and allowed to oscillate around their initial equilibrium position as a result of the binding/restoring force on the MNP from the microenvironment. The settling of the particles to a new static position is not necessary for the measurements reported in this study. A few oscillation cycles are sufficient for determining the resonant frequency of the sample. The static positions in these samples were measured to be stable over hundreds of milliseconds.
The requirement for the linearity of the viscoelastic material behavior is that the displacements induced be at most 0.2% of the length of the sample [31
]. In our case, the height of the samples was 5 mm, and therefore displacements of at most 10 μm would ensure a linear response and predict direct proportionality of natural frequencies with the square root of the elastic moduli. Moreover, in order to avoid confounding phase unwrapping, displacements did not exceed half the axial resolution of the OCT system, namely 1.5 μm. Therefore, the strength of the magnetic field was adjusted in the range of 100–600 Gauss for samples with different elasticities to ensure that this maximum displacement was not exceeded. The graph in shows the variation of the maximum phase change and displacement in a representative sample (with an RTV A:RTV B ratio of 10:1, a concentration of 2.5 mg/g of MNPs, 4 mg/g TiO2
and with a measured elastic modulus of 3.1 kPa) as the magnetic field strength is increased.
Fig. 3 Scatterer response to different magnetic field strengths. Direct measurements (points) of maximum change in unwrapped phase from an average scatterer, which are directly proportional to the average maximum displacements of the MNPs, as the electromagnet (more ...)
The natural frequency of oscillation of each sample was obtained from the time-resolved displacement of each scatterer, measured optically with the coherence ranging system. The displacement curves of the scatterers were fitted to the equation of motion of an underdamped oscillator with two frequency components according to Eq. (2)
is the displacement as a function of time, a1,2
are the amplitudes of the two frequency components, γ1,2
are the corresponding damping coefficients, f1,2
are the frequencies of oscillation, δ1,2
are arbitrary phases, and C
is a constant. The R-values of the curve fittings were all above 85%. The dominant natural frequencies of oscillation of the samples were plotted against the square root of the elastic modulus. Two frequency components were chosen in order to obtain a better fit of the displacement traces. These two frequency components are from an expected dominant longitudinal mode and possibly secondary harmonics or shear wave interference.
illustrates the normalized scatterer traces when the field was applied on three samples with different elastic moduli, as validated by indentation measurements. It is observed that the natural frequency of oscillation varies strongly with the elastic modulus and, as expected, stiffer samples exhibit higher frequencies. We note that some of the recorded displacement traces have secondary frequency components whose amplitudes are consistently smaller than those of the main frequency components.
Fig. 4 Normalized measured displacements from samples of different elastic moduli following a step (off-to-on) transition of the applied magnetic field. Three samples that span a wide range of elastic moduli (measured by indentation: 0.4 kPa [green], 6.4 kPa (more ...)