Mechanical Force Characterization of Living Cells based on Needle Deformation

The mechanical properties of cells play an important role in cell development and function. Therefore, measurement of cell mechanical properties is a fundamental and essential tool for cell research. In this article, a novel method to estimate the force exerted on a living cell is proposed based on glass needle deformation, which does not require additional physical sensors compared with other force‐sensing methods. The three‐dimensional (3D) spatial state of the needle is reconstructed, and the parameters of needle deflection are obtained based on a multi‐focus image fusion algorithm. The average reconstruction error of this algorithm is 0.94 µm. Based on the deformation of the needle, a mechanical model of needle deformation is established, and the model is calibrated using a constructed calibration system. At the range of 0–200 µN, the highest resolution is 0.002 µN and the lowest resolution is 5.3 µN. The proposed method can be used to estimate the force exerted by a needle on the surface of a living cell.


Introduction
The mechanical properties of cells influence cellular and subcellular functions, including cell adhesion, migration, polarization, and differentiation. [1][2][3] Robotic micromanipulation is a promising tool for cell research and has been widely used in biology and medicine, cell extraction and analysis, assisted reproduction, DOI: 10.1002/admi.202300293 single-cell injection, targeted drug transport, etc. [4] In cell manipulation, the end-effector or physical field interacts with the cell and exerts forces on it. The mechanical forces acting on cells may affect their functions and activities. Therefore, accurate measurement of the forces exerted on a living cell is essential for the biomechanical characterization of the cell, which is also critical for understanding physiological and pathological events at the single-cell level. [5,6] Owing to the complexity and fragility of biological structures, characterizing these tiny forces at the cellular and molecular levels is challenging.
Over the past few years, various microforce sensing methods have been developed at the cell level, for example, optical tweezers and vision-based, optical-based, capacitive, piezoresistive, and piezoelectric force sensors. [7][8][9][10][11][12] Optical tweezers are a non-contact force characterization method that measures the capture force on a cell by squeezing it with a particle controlled by a laser beam. [13] However, high-energy laser beams can cause photodamage to cells and affect their development. [14] Atomic force microscopy (AFM) is a typical optical force sensing device. [15][16][17] The light spot changes with the deformation of the load conduction medium and the circuit converts the detected change into an inductor output signal. It must be equipped with sophisticated transmission and receiving equipment, and the surface of the cantilever must be sufficiently reflective. The piezoresistive force sensor is based on the principle of the piezoelectric effect, in which the surface of a piezoelectric material generates an electric charge when it is squeezed. The electric charge is processed by an amplifier circuit and a measurement circuit to output the force, [12,18,19] and the measurement range is μN-mN. Because the sensor is sensitive to temperature, it is used at a constant temperature. Piezoresistive and capacitive sensors are devices that convert resistance or capacitance into highly sensitive electrical signals. [20,21] The resolution of the sensors is nN-μN, which is higher than that of piezoelectric sensors. However, the change in resistance is not easy to detect directly, and it is sensitive to the adhesive bonding quality, glue position, and temperature change. Capacitive force sensors are highly sensitive to noise and require complex electronics to filter it out. It is difficult to integrate these two sensors with microneedles, and the probe is fragile and easily damaged. These force sensors require additional physical components to measure changes in electrical Figure 1. Force estimation workflow. A multi-focus image fusion algorithm is adopted to reconstruct the 3D spatial state of the needle and obtain the parameters of the needle deflection. Based on the deformation of the needle, a mechanical model of the needle deformation is established and the model is calibrated by a constructed calibration system. The force exerted on the surface of a living cell is calculated using the established mechanical model.
parameters. Vision-based force sensors are cellular force estimation methods based on visual information that do not require additional physical components. Most of the available research has focused on the modeling of cell deformation, which relies heavily on the reliability of the cell model. [22,23] Owing to the complexity of intracellular organelles and mechanical properties, it is a challenging task to establish reliable mechanical models.
In response to these problems, we propose a method to characterize the force exerted on a cell based on glass needle deformation. The advantage of the proposed method is that it relies on the mechanical model of the needle, which is manufactured using a uniform and stable material. Compared to other forcesensing methods, this method does not require additional physical sensors. To realize the proposed method, a multi-focus image fusion algorithm is developed, which reconstructs the 3D spatial state of the glass needle and obtains the parameters of the needle deflection. Based on the deformation of the needle, a mechanical model of the needle is established and calibrated using the constructed calibration system. The proposed method is promising for estimating the force exerted by a needle on the surface of living cells.
The remainder of this article is organized as follows. The materials and methods are described in detail in Section 2. The ex-perimental results are presented in Section 3. Finally, Section 4 concludes the paper.

Experimental Section
The workflow of this study was shown in Figure 1. First, a multifocus image fusion algorithm was adopted, which reconstructs the 3D spatial state of the needle and obtains the parameters of needle deflection. Second, based on the deformation of the needle, a mechanical model of the needle deformation was established. The mechanical model was calibrated by a constructed calibration system and the force was calculated with the parameters obtained from the multi-focus images. The force exerted on the surface of a living cell was calculated using the established mechanical model.

Multi-Focus Image Fusion Algorithm.
The deformation of the needle can be treated as a 3D curve in space, which was difficult to capture using 2D microscopy. By taking advantage of the narrow field depth of the microscope (down to 1 μm), the 3D curve of the needle could be scanned by adjusting the focal length of the imaging system, as shown in Figure 2. A multi-focus image fusion algorithm was adopted to locate the focus area in the scanned images. [24] The model consists of two parts, DefocusNet and AiFNet, as shown in Figure 3.
DefocusNet learns a grayscale defocus map from a color input image. It takes a focal stack with an arbitrary size as input and estimates the corresponding defocus maps. The network uses an autoencoder as the basis and shares weights across all branches. Global pooling is used as a communication tool between separate branches. To compute the defocus map, a circle of confusion (CoC) is used as an evaluation index. When a light source passes through the camera lens, the light rays converge to form a focal point, which can be found on the image plane of the camera, as shown in   www.advancedsciencenews.com www.advmatinterfaces.de sharpness, which was equivalent to the amount of defocus. The equation used was as follows: where f was the focal length of the lens, S 1 was the focal distance, S 2 was the distance from the lens to the object, and N was the fnumber. The fnumber was the ratio of the focal length to the effective aperture diameter, essentially indicating the aperture size. All the pixel points in the image can be estimated using this metric. The input focus stack was processed using an autoencoder convolutional neural network (CNN). All the branches share the CNN weights. An arbitrary number of images were implemented as input by the global pooling layer by layer. A CNN network was used to learn and compare the focused regions with scattered regions. DefocusNet was trained using L2 loss without additional regularization.
To obtain a clear image of the needle, an all-in-focus (AiF) image needs to be obtained. Similar to the described DefocusNet, the AiF image is computed by attaching a CNN head, the AiFNet, in which all the pixels are properly sharpened. The DefocusNet model estimates a scattered focus map. The image sharpness level is determined for a given focal stack. By combining different image parts in the focal stack and their corresponding out-offocus maps, the final AiF image could be estimated. Specifically, AiFNet globally pools features from shared network branches, each of which processes the input images from the focal stack. AifNet consists of residual blocks. The main goal of AiFNet is to sharpen the input image and merge all the sharpened regions (represented by the defocused map) in a single AiF image. Furthermore, AiFNet has a second output that regresses the edge map to focus the training on sharp details. An edge map is a binary single-channel image that indicates the presence of an edge. The Canny edge detection method is used to precompute the true edge map on the all-focus image. It serves as an additional supervision for sharp details. The joystick communicates with the controller through a serial port, which can record data in the z-axis direction. The focusing plane of the microscope can be adjusted to clarify the different parts of the needle. The focused image is corresponded to the depth in the Z-axis direction to obtain a 3D state diagram of the needle.

Force Model
When the fine needle squeezes the cell, the cell is deformed and the needle bends, as shown in Figure 5. This was because the needle is squeezed by the axial force, which is greater than the critical yield force. Therefore, the needle does not maintain its original straight state. The needle can be considered a beam. Therefore, its force model can be simplified as shown in Figure 6. The relationship between the cross-section bending moment and curvature was where EI is the rotational stiffness and is the curvature. The curvature is calculated as follows: where is the angle at any section of the beam and S is the arc length along the deflection line of the beam. If the initial eccentricity of the axial force e is considered, the value of the Figure 6. Schematic of a simplified needle model. It is assumed that the needle deformation is uniform. The glass needle is bent by the axial force. The process of cells under compression can be simplified as a simply supported beam with one end fixed and one end moving. The mathematical relationship between the axial force and bending glass is established based on the simple supported beam theory. is the curvature, is the angle at any section on the beam, and S is the arc length along the deflection line of the beam. e is considered, and l is the length of the rod. is the angle between the glass needle and the horizontal direction when it is not bent. X 0 OY 0 is the Cartesian coordinate system, and XOY is the coordinate system at an angle with X 0 OY 0 .
www.advancedsciencenews.com www.advmatinterfaces.de section bending moment generated by the external load is where y is the deflection of the beam, and N is the axial force. The relationship between the arc length dS and the coordinates x and y is Combining the above equations, the control differential equation for the needle is Multiplying both sides of Equation 7 using d /dS, It can be obtained from Equation 8 that The integration of Equation 9 results in where C is the undetermined coefficient and k 2 = N/EI. Upon substitution of Equation 2 into 4, d /dS = −N(y − e)/EI could be obtained. Then, The Euler critical force form of the above equation is The Euler critical force is From Equations 10 to 13, the outcome is Let Thus, the outcome could be The integration of Equation 17 results in Upon substitution of Equation 5 into 14, the outcome is The positive and negative signs in the equation indicate that the pressure rod may be unstable in two different directions. Substituting Equations 15 and 16 into 20, Thus, where the value of C is solved using the first class of the fully elliptic integral, Equation 18. The elliptic equation and C can be expanded to obtain the second-order approximate solution of the midspan transverse deformation as follows: where A denotes where G denotes a component of N in the vertical direction. From Equation 24, the square of the deflection of the needle is proportional to the axial pressure. Because the needle is drawn at a high temperature, the tip diameter does not vary uniformly and is hollow inside. To establish the relationship between the force and needle deflection accurately, a numerical fitting method is used. Figure 7. Force calibration platform. The calibration system consists of a manipulator, an electronic balance, a microscope camera, a computer monitor, a needle holder, a joystick, and a robot controller. An electronic balance is used to measure force, and the microscope camera is used to record images of the needle bending. The needle is fixed to the needle holder, and the degree of needle bending is controlled by adjusting the height of the joystick.

Force Calibration Platform
To determine the relationship between axial force and needle deflection, a force model calibration platform is established. As shown in Figure 7, the calibration system consists of a manipulator (Transferman 4r, Eppendorf AG, Germany, resolution: step accuracy <20 nm, maximum working distance: ⩾20 mm, speed: 0-10000 μms −1 , angle adjuster: 0-90°), an electronic balance (Haozhan electronic scales, Rui'an Jinxun Trading Co., Ltd, resolution: 0.0001 g, working range: 0-220 g, working mode: electromagnetic force sensor), a microscope camera (E3ISPM20000KPA, Nanjing Dingcheng Co., Ltd, Frames per second (FPS)/resolution: 15/5440 × 3648, 50/2736 × 1824, 60/1824 × 1216, exposure time: 0.1ms-15s), a computer monitor (23.8″ IPS, Weixin Digital Franchise, Beijing), a needle holder, a joystick (TransferMan 4r, DualSpeed, Eppendorf AG, Germany, serial communication), and a robot controller. An electronic balance is used to measure the force, and a microscope camera is used to record images of the needle bending. The needle is fixed to the needle holder and the degree of needle bending is controlled by adjusting the height of the joystick. The calibration steps for this experiment are as follows: 1) Connection of devices. Connect all the devices in turn and ensure that all electrical connections and communication interfaces are working properly. 2) Calibration of the electronic balance. Use a 200 g weight to calibrate the balance. When using the electronic balance, it is important to ensure that the balance is not affected by other factors, for example, airflow from the air conditioner. 3) Force calibration. The needle is attached to the needle holder.
The angle button on the robotic arm is manually adjusted, which forms an angle between the needle and the horizontal surface of the electronic balance. The joystick is slowly adjusted to descend the needle slowly. The operation is stopped when the tip of the needle is in contact with the surface of the electronic balance as observed by the naked eye from the display screen. To ensure calibration, place the glass slide on the surface of the electronic balance and place rubber on the glass slide. The height of the needle is adjusted at a constant speed. When the electronic balance shows a digital change, the operation of the joystick is stopped, and the bending of the needle was photographed. The above experimental procedures are repeated 16 times. 4) Data processing and force model construction. The images are processed using image processing techniques to obtain the needle deflection parameters. The force data and the obtained parameters are fitted using the data processing software (OriginPro 2021 SR0) to obtain the force model.

Performance Analysis of Image Processing Algorithms
The dataset includes 1124 microscopic images at different focal lengths. All frameworks are trained end-to-end in a single NVIDIA RTX3060 GPU. The Pytorch 1.11.0 deep learning framework is deployed on the server running Ubuntu 18.04 with an 11th Gen Intel(R) Core (TM) i5-11400F @ 2.6G Hz. The initial learning rate is 0.0001. In the experiment, using a 2 μm glass needle with a length of 5 cm, we investigated the effects of the proposed algorithms in fusing multiple microscopic images with different focal lengths. To evaluate the performance of the multi-focus image fusion algorithm, the fusion error and FPS of the algorithm are analyzed. The average FPS of multi-focus image fusion algorithm is 3.88 ( Table 1). The image fusion results were compared with images labeled by a professional to calculate the image fusion error. The experiments on the four sets of images are shown in Figure 8, and the errors in the fusion results are shown in Figure 9. The maximum error of the algorithm was 3 μm in M3, whereas the maximum error of the other groups was less than 2.5 μm. The minimum error of the algorithm was 0 μm, and the average error (5-95 percentile range) was 0.94 μm. The data are evenly distributed between 0 and 2 μm. The median is between 0.5 and 1.25 μm. This shows that the image-fusion algorithm is reliable. M1-M4 represent the fusion errors of the four groups of images, which correspond to Figure 8a,b. The whiskers show the minimum and maximum recorded changes in the distance, the first and third quartile show the start and end of the box, and the band represents the median .

Force Model Calibration
In this study, a 2 μm glass needle with 5 cm length is selected for mechanical calibration. The values of the glass needle deflection and force at different angles were obtained, and the mechanical model was obtained using least-squares quadratic fitting. As shown in Figure 5, the X 0 -axis is assumed to be horizontal in the Cartesian coordinate system X O OY O , and represents the angle between the needle and the horizontal direction. The angles between the glass needle and the horizontal direction are 90°, 60°, and 45°, respectively. The angles between the glass needle and the horizontal direction were 90°, 60°, and 45°, respectively. The experiment was repeated thrice for each angle, and the experimental data are shown in Figure 10. The three sets of experimental data have the same trend, which indicates that the data are reproducible. According to Equations 24 and 25, the numerical fit between the glass needle deflection x and axial force F is shown in Figure 11. The force model is  where F is the force exerted by the needle on the surface cell and x is the deflection of the needle. The coefficient of determination R 2 is 0.9787. The root mean square error (RMSE) of the model is 7.26μN. The sensor resolution or measurement resolution is the smallest change that can be detected in the quantity that it is being measured. [25] Based on the resolution definition, we measured the force resolution of the proposed method at 0-200 μN. The relationship between the sensor resolution and measurement range is described in Figure 11c. In the range of 0-200 μN, the Figure 10. Needle deformation at different angles with the corresponding force data. The force data here is the recorded value of the electronic balance, representing the vertical force G, which is the component of the axial force N applied to the rod. There is an angle between G and N, and is 90°, 60°, and 45°, respectively. highest resolution is 0.002 μN/μm and the lowest resolution is 5.3 μN/μm. For convenience of application, we measured the mean resolution corresponding to different measurement ranges, as presented in Table 2.

Force Characterization Experiments
The proposed method can be used for cell-force characterization based on needle deflection. Rice microspores (with a diameter of 50 μm) and zebrafish embryos (with a diameter of 1000 μm) were selected as the experimental objects, as these are two typical cell scales and easily available materials. The aforementioned model was applied to assess the force exerted on the tested cell. While puncturing the cells, the microscope focus knob was adjusted at a certain speed and the entire puncture process was recorded with a camera. A multi-focus image algorithm was used to calculate the deflection of the glass needle. The puncture force of the cells was calculated using a force model. The experimental platform for the force characterization is shown in Figure 12.
The results of the force characterization experiments are shown in Figure 13. At this time, the force exerted by the needle on the surface of the rice microspore was 25.37 μN with a deflection of 50.50 μm. The force exerted by the needle on the surface of the zebrafish embryos was 132.60 μN with a deflection of 106.50 μm.

Discussion
Various cell microforce sensors have been developed to characterize the mechanical properties of cells, as presented in Table 3. Wei et al. [26,27] designed two types of sensors, piezoelectric sensors and piezoresistive sensors, which are large-range sensors. However, the sensor resolution (0.65 and 0.80 mN) is low; therefore, these sensors are suitable for cells and microorganisms above 1 mm. Muntwyler et al. [28] proposed a capacitive microforce sensor with a regulable force range (from ±20 to Wei et al. [26] Piezoresistive sensor 0-16.5 mN 0. mN Wei et al. [27] Piezoelectric sensor 0-100 mN 0.80 mN Muntwyler. [28] Capacitive sensor ±200 μN Best resolution 30nN Zhang et al. [29] Optical-based force sensor 0-60 μN Less than 1 μN Tan et al. [30] Vision-based force sensor 0 -  ±200 μN) through different readout electronics settings for wider applications. However, such sensor probes are fragile, susceptible to damage, and difficult to integrate into microneedles for microinjection. Vision-based force sensors are affected by inevitable parameter uncertainty in the dynamic cell model. [30] Moreover, vision-based force sensors are highly dependent on the availability of a suitable cell model. The proposed method does not rely on complex devices or an accurate cell mechanical model, and it can be integrated with a microneedle for microinjection. In the measurement range of 0-200 μN, the sensor has a minimum resolution of 5.3 μN/μm, which has certain superiority.

Conclusion
In this article, a novel method to estimate the exerted force is proposed based on glass needle deformation, which is important for biomechanical characterization of live cells. The deflection of the glass needle in space is obtained based on a multi-focus image fusion algorithm, and the force exerted on the surface of the living cell was calculated using the established mechanical model. The proposed method does not require any additional physical components.
The average reconstruction error of the multi-focus fusion algorithm is 0.94 μm, and the reconstruction speed is 3.87 FPS for  an image with a size of 1920×1072. The RMSE of the mechanical model is 7.26 μN. In the cell force characterization experiment, the force exerted by the needle on the surface of the rice microspore is 25.37 μN with a deflection of 50.50 μm. The force exerted by the needle on the surface of the zebrafish embryos is 132.60 μN with a deflection of 106.50 μm. The proposed method shows promise in live cell biomechanical characterization.