INTRACELLULAR Ca2+ waves play an integral part in the process of excitation-contraction coupling within mammalian cardiomyocytes. Ca2+ waves might spontaneously arise from a triggering pulse of Ca2+ released from the sarcoplasmic reticulum, which under certain circumstances propagates throughout the cell (1–3). The qualitative and quantitative characterization of Ca2+ waves, including amplitude, frequency, and propagation velocity, is thus a central issue for understanding both normal and pathological contractile cell behavior, such as the correlation of spontaneous Ca2+ waves with arrhythmic electrical activity (4–6). Despite obvious limitations, such analyses are usually conducted on isolated single cardiac cells. Indeed, within the myocardium, the cardiomyocytes are arranged in a complex network. Thus, correlating accurately forces and length changes with intracellular Ca2+ waves is rather difficult. Monitoring Ca2+ waves has then been made possible by using fast confocal microscopy techniques, which image the intensity variation of calcium fluorescent probes such as Fluo-3 (7, 8). In addition, such probes have been proven to be highly valuable for analyzing the dynamic of intracellular Ca2+ components, like ryanodine receptors (9, 10).
On the other hand, the direct quantification of sarcomere mechanical status becomes highly valuable to get insights into the regulation of cardiac muscle contraction mediated by the mechanosensitivity of biochemical processes or molecular interactions, which depend on the sarcomere length or on actin-myosin filaments spacing. Sarcomeres length measurements are usually based on laser diffraction techniques (11), but this approach is not well suited for the analysis of the spatio-temporal variation of the sarcomeres length at the cell level, especially when the dynamic of the cardiomyocyte contraction wave has to be characterized.
To overcome these limitations, we developed an optical flow approach aiming at characterizing the spatio-temporal variation of intracellular strain fields associated with sarcomeres contraction–relaxation over the course of cardiomyocyte contraction. Even though optical flow methods are rather widely applied in several engineering or bioengineering fields, only a few attempts have yet been made to apply them to single cell dynamics analysis (12–14), mostly because animal cells are highly deformable objects. Among the rather large spectrum of optical flow methods implementations (15, 16), Image Correlation Method (ICM) (17–19) seemed to us a quite relevant approach for characterizing intracellular strain fields during cardiomyocyte contraction. Indeed, the highly structured and anisotropic organization of the cell sarcomeres defines a very precise and repetitive spatial pattern, which may be used in a rather discriminating way by ICM when looking for a similarity between two successive patches of the region of interest (ROI) within cell images. Thus, we were expecting that such a priori optimality can be quantitatively assessed by developing an image-processing algorithm that takes benefit of the specific grey level patterns generated by phase contrast observation of contracting cardiac cells.
In this article, we propose and validate a complete procedure which provides an accurate estimation of the 2D displacement fields at any cardiomyocyte location from the analysis of time-lapse sequences of single cell contraction recorded using phase contrast and differential interface contrast (DIC) videomicroscopy. An essential step in this procedure is to find the displacement field, which optimizes the prediction of the light intensity pattern which will be observed when considering two successive images of the sequence. The first part of the article focuses on the method implementation. Then, our method reliability and accuracy is assessed by analyzing different sets of synthesized pair of images, constructed from known intensity displacement fields. The second part of the article is devoted to the analysis of experimental time-lapse sequences showing different spatio-temporal patterns of spontaneously contracting adult rat cardiomyocytes. For each considered experiment, the spatial distribution of intracellular strain fields has been characterized at several times during the cardiomyocyte contraction. In the discussion section, the robustness of our approach is discussed by considering less textured contracting objects. For this purpose, image sequences of spontaneously contracting neonatal cardiomyocytes have been analyzed, since at this stage of development, the cytoskeleton of these cells lacks the regular sarcomere organization present in adult cells.
MATERIALS AND METHODS
Adult rat cardiomyocytes were kindly provided by the LBFA laboratory (Joseph Fourier University, Grenoble). Briefly, cells were isolated from rat heart ventricles by enzymatic dissociation as described in Ref.20.
Time-Lapse Video Microscopy
The spontaneous and periodic contractions of the isolated adult rat cardiomyocytes were recorded by DIC and phase contrast time-lapse videomicroscopy using an imaging workstation composed of an inverted microscope (Zeiss Achrostigmat Axiovert 135) equipped with a 10× objective (Zeiss Acroplan), automated shutters (Uniblitz) and a CCD camera (CoolSNAP HQ, Roper Scientific). Image acquisition was controlled via a computer using a dedicated software (Metaview, Roper Scientific). Microscope calibration was performed using a micrometric slide (PRESS-PRO21). The size of the images we analyzed was 442 × 161 pixels. Images were taken every 110 ms at room temperature.
Image Correlation Analysis and Displacement Field Computation
An ICM was used to compute the displacement field w(u,v) between a pair of successive images taken at time t and t+Δt, and labeled for convenience as Image 1 and Image 2, respectively. Let us consider a small window area Wind1 in Image 1 (Fig. 1). Then, the aim of the algorithm is to find the matching window Wind2 in Image 2, i.e., the position to which Wind1 has moved during the time interval Δt. For this purpose, Wind1 is compared with windows belonging to the same neighborhood Neigh2 in Image 2 (Fig. 1). Neigh2 is also called the region of interest (ROI). For each window Wind2, a similarity measure is computed. The window exhibiting the highest similarity, i.e. the best correlation with Wind1, is then retained as the matching window Wind2*.
More formally, let us denote IW1 the illumination intensity matrix of Wind1, and IW2 the illumination intensity matrix of any current window Wind2 of the ROI, respectively, with
where any integer It(i,j) defines the reference grey level value at time t of the pixel located at (i,j) in Image 1, while It+Δt(i+u, j+v) defines the corresponding current grey level value of the pixel located at (i+u, j+v) in Image 2, corresponding to time t+Δt.
For any window Wind1, we are looking for an optimum window Wind2* such that
and (u*,v*) are the two unknown components of the optimum displacement vector w*(u*,v*) with respect to the chosen similarity or discrete cross-correlation function C(.,.).
Among all different similarity functions proposed, we chose a variance-normalized cross-correlation function CV(.,.), since such weighted criterion has been shown to be more robust (21, 22).
Considering two matrices IW1 and IW2 with dimension (N,M), the similarity function CV(.,.) is computed for each successive pair of the image sequence as
where It and It + Δt represent the respective means of It and It+Δt defined as
The algorithmic strategy used for computing w(u,v) follows different guidelines. A major point is to limit the size of the ROI in order to optimize the speed, robustness, and accuracy of our algorithm. Indeed, dealing with small displacements requires considering neighborhood of small size and imposes large computational times, but choosing too large neighborhoods would likely lead to non-meaningful displacement solutions, as illustrated below.
Optimization of the ROI
Detection of the matching window Wind2* must take care of both the displacement amplitude and of the background noise in the illumination maps, this latter originating from different sources. These include fluctuations of illumination intensity with object displacement (departure from illumination conservation assumption), as well as fluctuations of the focusing plane in the case of a contracting 3D object like a cardiac cell. This problem has been handled by considering an optimal size of the ROI in association with an optimum value of the similarity function Cv(IW1,IW2), according to the following procedure: taking a square window Wind1 of size (si×si) and centered at (X) in Image 1, we defined a ROI centered at (X, Y) in Image 2 with size (si + (ka × si) × (si + (ka × si))), where ka is a dilation scale-factor (ka ≥ 1). We started the search of the optimum displacement vector w* by considering a ROI two times larger than Wind1 (ka = 1). This gives a first value δ1 of Cv(IW1,IW2). Then, the value of the scaling factor ka is increased by an integer value of 1, which generates increasing values δka that tend to plateau toward the best reachable correlation value δopt of Cv(IW1,IW2), with δopt ≤ 1. The larger is the value of δopt, the more accurate is the detection of the matching window Wind2*. This point is illustrated in Figure 3 introduced below. A global estimation of a correlation threshold δ was obtained by averaging all the values δopt over Image 1.
Construction of Pairs of Test Images
To test the accuracy and precision of the method, different pairs of test images were constructed by applying specific and known spatial displacement fields on a textured square object with size [0,1] × [0,1] [Image 1, Fig. 2(A)] to get corresponding deformed objects [Image 2, Figs. 2(B) and 2(C)]. The size of Image 1 is 256 × 256 pixels. Meshing the square with 2D triangular elements defines a set of nodes where known displacements have been applied.
Case 2: The second set of test images was obtained by imposing to the textured squared object a more complex displacement field, defined by u(x,y) = 0.1 y and v(x,y) = 0.1 x [Fig. 2(C)].
Derivation of the Strain Field Associated with the ICM Computed Displacement Field
For each matching window Wind2*, centered at (X, Y) and associated with a reference window Wind1 centered at (X, Y) at time t, we derived the displacement field w between times t and t+Δt as:
To describe the transformation of any window, located at (k,l), between times t and t+Δt, we used the deformation gradient tensor F given by
The relation between the Green-Lagrange strain tensor E and the deformation gradient tensor is given from continuum mechanics by (23):
where Id is the identity tensor and (.)T denotes the tensor transposition.
The strain field with regard to the initial (or undeformed) configuration at t0, i.e. for generally large deformations, is given at any time t by the Green-Lagrange strain tensor strain E computed from the composition of the successive deformation gradient tensors:
from which we obtained the normal strains components Exx,t, Eyy,t and shear strain component Exy,t, where the subscript t0 has been omitted to simplify notations.
When deformations are computed between two consecutive images, small deformations analysis was performed. We then evaluated the associated linearized normal εxx,t, εyy,t, and shear strains εxy,t between two successive images as
These derivatives were computed numerically using the following finite differences approximations
Accuracy, Robustness, and Optimal Window Size of the Optical Flow Algorithm
Both pairs of test images were analyzed using our optical flow algorithm. Accuracy and robustness were assessed from comparison of the computed displacement vector fields with the corresponding known numerical or analytical solutions [case 1, Fig. 2(B); case 2, Fig. 2(E)]. Figures 2(D) and 2(G) present the known displacement field imposed for synthesis of each Image2 of the test images pairs. In both cases, the present algorithm successfully reconstructed the corresponding displacement maps [Figs. 2(C)–2(F)], with a maximum error of 15% on the computed displacements.
These test images have been also used to determine the optimal size of the research window. When considering the first pair of test images [Figs. 2(A) and 2(B)], our computations showed (Fig. 3) that the optimum size for square window Wind1 is close to (10 × 10) pixels, while the best correlation threshold δ has been obtained with a dilation scale-factor ka = 3, corresponding to a ROI in Image 2 with size of (40 × 40) pixels.
Quantification of Cardiomyocyte Shortening from Computed Contraction Waves
This section presents the analysis of two typical spontaneous contractions of isolated adult rat cardiomyocytes. It is known that cell contractions triggered by calcium sparks can appear spontaneously, i.e. in the absence of external stimulation, and randomly within the cell when cardiomyocytes are cultured in vitro (24).
We will focus here on two different contraction patterns that we observed in our in vitro experiments. In the first contraction sequence (seq1), the contraction wave started from the left side of the cell and propagated up to the right boundary [Figs. 4(1A)–4(1E)]. Interestingly, the same cardiomyocyte later exhibited a rather different contraction pattern (seq2), the contraction being then initiated at the center of the cell (Figs. 5(2A)–5(2E)]. In this contraction pattern, two strain waves now propagated in opposite directions, toward the cardiomyocyte extremities. A measured time lag of 18.34 s separated the two cell contractions.
Global Quantification of Single Cardiomyocyte Shortening
Figure 6A shows the dynamics of the cardiomyocyte shortening recorded by sequences seq1 and seq2, respectively. The cell shortening was measured in three different ways. First, a global evaluation of the cell length was performed, for both seq1 and seq2, by finding the length of the smallest rectangle surrounding the cardiomyocyte (Fig. 6(A), solid lines). Alternatively, cardiomyocyte shortening in seq1 was computed from the integration, along a longitudinal middle-cell cross-section, of the local intracellular strains derived from the ICM approach (Fig. 6(A) left, square markers). Such one-dimensional integration from local to global cell length is not valid for seq2, since then the cardiomyocyte underwent significant bending during contraction. In this case, the quantification of the cell shortening from the ICM approach was restricted to the consideration of windows Wind1 located at both cell borders (Fig. 6(A) right, circular markers).
During the first contraction (seq1), the cell contraction/relaxation took place within 2.05 s and represents 6% (7.13 μm) of the maximum cell length (112.05 μm). The contraction/relaxation phase was shorter (1.21 s) and two times larger (13.69 μm) in seq2, since then the two cell sides contracted simultaneously.
The times needed to reach the maximum amplitude of contraction were 0.57 s and 0.33 s in seq1 and seq2 respectively. In seq1, contraction/relaxation began on the left side. During the left side relaxation contraction/relaxation occurs in right cell borders. In addition, we checked that the image analysis results given by moving windows located at the adult cardiomyocyte extremities gave realistic and accurate antagonistic displacement profiles for both cardiomyocyte extremities [Figs. 6(B) and 6(C)]. These windows exhibited the same profiles as those reported in previous studies using attached carbon fibers (25, 26).
The agreement between globally based and locally based measurements methods is quite satisfactory, and thus illustrates that an accurate and combined analysis of the cell contraction and associated intracellular displacement fields is provided by the ICM-based analysis, as detailed below.
Characterization of the Associated Intracellular Contraction Patterns
The strain waves, which propagate through the contracting cardiomyocyte, have been tracked for the two contractile behaviors recorded in the time-lapse sequences seq1 and seq2. Thanks to our optical flow analysis, the local displacement field can be computed in any part of the cardiomyocyte, thus giving access to intracellular mechanical features of intracellular contraction/relaxation at the sarcomere level. In this analysis, the size of Wind1 is 10 × 10 pixels, while a ROI-scale factor ka =2 and a ROI of 30 × 30 pixels were chosen.
In the first videomicroscopy sequence, the computed motion field revealed four main steps during cell contraction. The first one was characterized by the cell contraction starting on the left cell side, the right cell side remaining at rest during 440 ms [Fig. 4(1A)]. In the second stage (220 ms), sarcomeres contractions occurred on both sides of the propagating front [Fig. 4(1C)]. In the third stage, sarcomere relaxation began on the left cell side, while contraction still proceeded on the right side, during 330 ms [Fig. 4(1D)]. Finally, complete relaxation took place on the right cell side (550 ms) [Fig. 4(1E)].
Analysis of the second time-lapse sequence (seq2) highlights two main phases during cardiomyocyte contraction/relaxation. During the first phase, an asymmetric distribution of intracellular stresses at the cell center lifted the cell extremities upwards (550 ms) [Figs. 5(2A) and 5(2B)]. However, this twisted motion was immediately followed by a reverse torque, which tended to shift the cell back, probably as a result of homogenization of calcium concentrations within the cell (550 ms) [Figs. 5(2C) and 5(2D)].
Analysis of Intracellular Strains
A main benefit of our approach is its ability to provide a quantification of the intracellular strains everywhere within the cardiac cell. This section presents the estimated strain fields we derived from the displacement fields computed by the optical flow method, using Eq. (10).
For two observed typical contractile behaviors of isolated cardiomyocytes, we reported in Figure 7 the components of the 2D strain tensor, namely (i) horizontal strain Exx,t along the cell length, i.e. in the sarcomere direction and (ii) transverse strain Eyy,t along the cell width. Local cell stretching is characterized by a positive strain, while a negative strain indicates a cell twitch.
When contraction propagated from left to right (seq1), the wave front was preceded by negative values of strains Exx,t, Eyy,t [Figs. 7(1A)–7(1C)] and followed by positive strain values. Absolute strains values were in the range Exx,t = 12–15%, with Eyy,t = 8%.
In the second videomicroscopy sequence seq2, the contraction wave, which emerges in the middle of the cardiomyocyte, was associated with negative strains along both propagating directions. This contraction traveled along the cell horizontally and vertically [Figs. 7(2A)–7(2C)]. Immediately after this contraction wave, a dilation wave front started from the cell center and was characterized by positive strains. Corresponding strains values were Exx,t = 15–20% and Eyy,t =7–10% for both contraction waves.
Let us notice that the shear strain value Exy,t was not negligible, with Exy,t ∼10% for both sequences. In seq1, the wave propagation occurred mainly along the sarcomere direction and a significant shear strain developed mostly in the cell corner during cell contraction. In seq2, shear strains were more homogeneously distributed and appeared all along the cell (data not shown).
Evaluation of the Algorithm Performances When Analyzing Time-Lapse Sequences Including Less Textured Images of the Cardiac Cell Contraction
Neonatal cardiomyocytes present a less organized intracellular architecture than adult cardiac myocytes, and thus provide an interesting application field to evaluate our approach in the case of less textured images. Thus, we analyzed time-lapse sequences of contracting rat neonatal cardiomyocytes, isolated from the ventricles by enzymatic dissociation (27), and computed the strain fields generated during cell contraction. The cell contraction occurred with higher frequency, the contraction motion being then mainly centripetal (Fig. 8). The strains values have been still computed accurately by the optical flow method, with absolute values in the range of 1–3% for εxx,t and εyy,t, with shear strains εxy,t in the order of ∼2%. This indicates that adult rat cardiomyocytes generate intracellular strains two to three times larger than neonatal cells. These results compares quite well with the 3-fold increase in force normalized to the cross-sectional area (FCSA) reported for murine skinned cardiomyocytes (28) and skinned trabeculae from rat ventricles (29).
In this study, a correlation-based optical flow method has been developed to analyze time-lapse sequences of contracting cardiomyocytes and to quantify originally spatio-temporal variations of intracellular strains generated at a sarcomere scale. The proposed image processing method allowed a precise tracking of the strain wave traveling along the cardiac cell by analyzing the correlation of grey level patterns between consecutive pairs of images.
While belonging to a class of image processing methods rather widely used for analyzing objects motions from image sequences (16), few applications have been reported in the context of cell dynamics to date (12–14). Indeed, the association of the usually limited contrast of videomicroscopy images with large cell deformations defines a challenging framework for implementation of optical flow methods. Thus, in order to obtain a 2D vector representation of image pixels displacements, regularization conditions are usually added to constrain the search for the optimum displacement field solution. The method developed here presents substantial improvements compared with the method used in other studies (13, 30), since it avoids the need for choosing regularization constraints. In fact, our method takes benefit of the landmarks provided by the sarcomeres structural organization, and thus provides a precise characterization of cardiomyocyte contraction and deformation without requiring the coupling of phase contrast observation with intracellular calcium imaging by fluorescent probes. In addition, there is no need for linking specific markers, such as microbeads, on the cell surface to track complex cell distortions (31–33).
Our approach also differs from other optical flow approaches developed for analyzing oscillatory dynamics of cell protrusions (12) or tissue contractions (34), which model the flow by a linear transformation. As a result, one gets a compact and global measurement of the biological object deformation from the time evolution of motion descriptors associated with the different types of singular points, characterizing the velocity field computed between two consecutive images (19).
In the present study, we were looking for a simple and efficient way to link observations made at the whole-cell level to dynamical features exhibited at the sarcomere level. The proposed approach appears rather powerful and satisfactory since the integration of the displacement fields computed at this sub-cellular scale are in excellent agreement with the global measurements of cardiomyocyte shortening obtained by tracking the motion of both cell extremities. As a consequence, we were able to accurately identify the amplitude and spatio-temporal variations of the strains wave that characterize successive sarcomere contraction/relaxation.
The accuracy of the method already insures that displacements of a few microns can be reliably detected. However, our algorithm can still be extended to allow sub-pixel accuracy. The temporal resolution of the method obviously depends on the sampling rate of image acquisition, and can take benefit of the ultra-fast cameras already used to image intracellular calcium dynamics.
Finally, we would like to discuss possible implications of our image analysis approach. First, we can notice that the propagation of the negative longitudinal strains Exx,t (Fig. 9), corresponding to a localized intracellular compression, can be viewed as a mirror image of the underlying propagating peak of cytosolic calcium that drives the cell-contracting twitch (35).
Second, we might wonder if such similarity-based optical flow approach still remains reliable when analyzing mechanical deformation of cells, which do not exhibit regular and organized grey-level intensity patterns when observed by phase contrast or DIC microscopy. As a reply, we considered the contraction of cardiomyocytes taken at an earlier developmental stage. Although rat neonatal cardiomyocytes lack the well-organized band-like arrangement observable in adult cells, our ICM approach still successfully quantified the rather complicated cell dynamics, including phases of centripetal contraction and generation of localized intracellular shear strain.
In conclusion, the proposed approach offers significant improvements to the classical ways of quantifying single cardiomyocyte deformations dynamics and using either force transducers, lasers diffraction techniques or fluorescent probes. By providing a direct and noninvasive access to intracellular strain at the sarcomere scale, this image processing method provides information on both active and passive mechanical properties of cardiac cell in a noninvasive way, i.e. without requiring the use of external probes, as it is the case in micromanipulation experiments like MTC experiments (36). More fundamentally, this optical flow method might also help to investigate the influence of mechanosensitive biochemical processes, depending on sarcomere stretching or interfilament spacing (37), which control the contractility and efficiency of cardiomyocyte contraction (38).
The authors thank Drs. R.S. Chadwik (NINDS-National Institutes of Health) and Dr. C. Hongxue (Northwestern University) for their useful comments, as well as Drs. N. Béreaud, J. Olivares (LBFA, Univ. J. Fourier), and M. Trifa (CNRS, TIMC) for their technical support.