Machine learning method for processing forcedistance curves in force microscopy

Abstract

The present invention relates to a method for processing force versus distance curves obtained by means of a force microscope by applying a supervised machine learning method based on functional multilayer perceptron neural networks. The method allows for rapid processing of thousands of experimental curves and eliminates errors associated with the nonlinearity of the AFM tip displacement signal. This step allows the accurate prediction of various mechanical and nanomechanical properties such as Young's modulus, compressive modulus or fluidity coefficient. Likewise, the method of the invention 0 facilitates the measurement of nanomechanical properties of materials in real time.

Description

[0001] MACHINE LEARNING METHOD FOR PROCESSING FORCE VERSUS DISTANCE CURVES IN FORCE MICROSCOPY

[0002] OBJECT OF THE INVENTION

[0003] The present invention falls within the scope of the development of force microscopy methods, these methods being applied to samples of polymers, biological cells or tissues, whether in a liquid or gas medium. The present invention has application in fields such as microscopy, clinical research, mechano-biology, cell biology, nanomedicine or the development of advanced polymers.

[0004] STATE OF THE ART

[0005] Force microscopy (AFM) is the most used technique for the characterisation of mechanical properties at the nanoscale (R. Garcia, Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications, Chem. Soc. Rep. 49, 5850-5884 (2020); G. Stan, S. W. King, Atomic force microscopy for nanoscale mechanical property characterization, J. Vac. Sci. TechnoL B 38, 060801 (2020); M. Krieg, G. Flaschner, D. Alsteens, B. M. Gaub, W.H. Roos, G.J.L. Wuite, H. E Gaub, C. Gerber, Y. F Dufrene, D. J Muller, Atomic force microscopy-based mechanobiology, Nature Reviews Physics 1 , 41 -57 (2019)). As a consequence of their high spatial resolution (lateral and vertical), force microscopes (AFM) have been introduced both in research laboratories and in quality control departments in various industrial sectors such as microelectronics, polymers, food or pharmacy.

[0006] One of the singularities of the force microscope is its capability for providing measurements and maps of topographic and mechanical properties of a material. Among the mechanical properties are adhesion forces, Young's modulus, compressive modulus, coefficient of viscosity or parameters related to the compliance of the material under study. Thus, a wide variety of AFM-based methods have been developed, among them, methods based on measurements of force versus displacement curves (H.J. Butt, B. Capella, M. Kappl, “Force Measurements with the Atomic Force Microscope: Technique, Interpretation and Applications” Surf. Sci. Rep. 59 1 -152 (2005); R. Garcia, Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications, Chem. Soc. Rep. 49, 5850-5884 (2020)).

[0007] The force versus distance curve (FDC) acquisition method, also known as AFM nanoindentation, is widely used for characterising the nanomechanical properties of polymers, biomolecules, cells, hybrid materials, and nanostructures. This method requires the acquisition of many FDCs to obtain accurate values of the mechanical properties of materials, in particular, for very soft samples such as biological cells or tissues. This process involves high processing times (hours) which limits its applications in fields such as biomedicine or clinical analysis.

[0008] The present invention is based on the incorporation of a machine learning method for processing force versus distance curves and predicting the nanomechanical properties without model fitting. So that the processing time is reduced and the accuracy of the values of the mechanical properties is increased.

[0009] Recently, several machine learning methods have been proposed for processing data obtained by AFM. For example, several contributions have proposed methods for classifying images of biomolecules, in particular cells, and associate the images with different types of physiological states and / or pathologies (S. Prasad, A. Rankine, T. Prasad, P. Song, M. E. Dokukin, N. Makarova, V. Backman, and I. Sokolov, Atomic Force Microscopy Detects the Difference in Cancer Cells of Different Neoplastic Aggressiveness via Machine Learning, Adv. NanoBiomed Res. 1 , 2000116 (2021 ); K. Ito, Y. Ogawa, K. Yokota, S. Matsumura, T. Minamisawa, K. Suga, K. Shiba, Y. Kimura, A. Hirano-lwata, Y. Takamura, T. Ogino, Host Cell Prediction of Exosomes Using Morphological Features on Solid Surfaces Analyzed by Machine Learning, J. Phys. Chem. B 122, 6224-6235 (2018). Scientific activity in machine learning methods dedicated to classifying AFM images has been summarised by several authors (I. Azuri, I. Rosenhek-Goldiani, N. Regev-Rudzki, G. Fantner, S. R. Cohen, The role of convolutional neural networks in scanning probe microscopy: a review, Beilstein J. NanotechnoL 12, 878 (2021 )).

[0010] Likewise, a machine learning method has been proposed to define the most appropriate elastic contact mechanics model to simulate the deformation of a cell to the force exerted by the AFM (L.T.P. Nguyen, B.H. Liu, Machine learning framework for determination of elastic modulus without contact model fitting, European Journal of Mechanics I A Solids 94, 104579 (2022).

[0011] Finally, several approaches based on machine learning methods have been proposed to determine mechanical properties based on AFM experiments. For example, a comparison has been made between several regressors, including the one known as 'Gaussian process regression' to define the elastic modulus of cells based on force versus distance curves (L.T.P. Nguyen, B. H. Liu, Machine learning framework for determination of elastic modulus without contact model fitting, Inter. J. Solids and Structures 256, 11 1976 (2022)). Machine learning methods have also been applied to estimate the force exerted on polymers using dynamic modes such as tapping mode AFM (A. Chandrashekar, P. Belardinelli, M. A. Bessa, U. Staufer, F. Alijani, Quantifying nanoscale forces using machine learning in dynamic atomic force microscopy, Nanoscale Adv. 4, 2134 (2022)).

[0012] Previous methods devoted to classifying images of biomolecules do not provide information about the nanomechanical properties thereof. Moreover, the methods that do contemplate the characterisation of mechanical properties from the FDC curves only provide the elastic modulus. It is known that the mechanical response of a cell is viscoelastic (R. Garcia, Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications, Chem. Soc. Rep. 49, 5850-5884 (2020)). Furthermore, the fact that the indentation signal may have a non-linear behavior with respect to the displacement of the piezoelectric and that this may be carried out at different speeds or frequencies was not contemplated.

[0013] Figure 1 shows a scanning electron microscopy image of a cantilever used in force microscopy to measure cells (model FastscanJD, Nano sensors). Figure 2 shows an example of an experimental curve of force versus distance (a) and the same curve versus time (b). The tip displacement signal is shown in panel c. The force curves have been measured with an AFM on a living HeLa cell.

[0014] In general, in a nano-indentation measurement, the nanomechanical properties of a material, for example, a cell, are determined by fitting a force-distance curve to a contact mechanics model. Since cells are viscoelastic materials, the contact mechanics model to describe them will have to be viscoelastic. It has been shown that the model known as 'power-law rheology' (PLR) is the most efficient model to simultaneously simulate the approach and retraction sections of a FDC measured on a cell (PD Garcia, CR Guerrero, R Garcia, Nano rheology of living cells measured by AFM-based force-distance curves, Nanoscale 12 (16), 9133- 9143 (2020); J.G. Sanchez, F.M. Espinosa, R. Miguez, R. Garcia, The viscoelasticity of adherent cells follows a single power-law with distinct local variations within a single cell and across cell lines, Nanoscale 13, 16339-16348 (2021 ).

[0015] A PLR model is considered here including the bottom-effect correction to minimise the influence of the Young's modulus of the rigid substrate on the nanomechanical properties of the material under study (P.D. Garcia, R. Garcia, Determination of the viscoelastic properties of a single cell cultured on a rigid support by force microscopy, Nanoscale 10, 19799-19809 (2018). In this model and for the case of a semi-infinite sample, the force as a function of the indentation / is calculated using: wherein - t' is the viscoelastic relaxation function of the material, and which in the event of a conical tip and a linear indentation profile with speed (v) transforms into:

[0016] Wherein a is a coefficient that depends on the geometry of the indentation and the Poisson's modulus of the material, [3 is a coefficient that depends on the geometry of the indentation, tmax is the time for which the indentation reaches its maximum value and ti(t) is the solution to determine the contact area during tip’s retraction, obtained by applying Ting's condition (T.C.T. Ting. The Contact Stresses Between a Rigid Indenter and a Viscoelastic Half-Space. J. AppL Meeh. 33, 845-854 (1996).

[0017] The relaxation function ( / ( / - 0 for the PLR model takes the form: wherein r is the Euler gamma function, to is a reference time, t0= 1 s is usually taken to facilitate calculations; Eo is the elastic modulus of the material at time t0, a scaling factor with units of force divided by area, and y is the fluidity coefficient that varies between 0 and 1 . For an elastic solid of Young’s modulus Eoit takes the value y=0 and for a Newtonian viscous liquid it takes the value y= 1 , with viscosity ie= Eoto.

[0018] For finite-thickness materials, i.e., with a thickness h the force equations transform into: wherein aj is a coefficient that depends on the geometry of the tip, the Poisson's coefficient of the material (v=0.5 for biological cells and tissues) and the thickness of the cell (h). Moreover, / 3y just depends on the geometry of the tip.

[0019] The coefficients for a conical tip are detailed in table 1 :

[0020] Table 1 shows the coefficients of the bottom-effect correction for a conical tip of

[0021] > tan 0 semi-angle 0. The expansion term e is defined ase -h and the proportionality constant i

[0022] When the tip of the AFM is spherical, a parabolic approximation can be used when the indentation is small compared to the radius of the probe. In this case, the coefficients of the bottom effect correction are the following: Table 2

[0023] Where the expansion term is b = — with R the radius of the sphere and the proportionality constant c is defined as c This model can be solved analytically only for specific indentation profiles, such as linear indentation.

[0024] Figure 3 shows force-distance curves obtained according to the PLR model for a conical tip with semi-angle 0 = 18°. The figure shows curves with different values of the fluidity coefficient and fixed compressive modulus Eo = 1000 Pa (a) and curves with different values of the compressive modulus and a fixed fluidity coefficient y = 0.2 (b).

[0025] It is difficult to find analytical solutions for the integral equations E.5 and E.6. Specifically, it has only been possible to obtain them for triangular type displacement signals. Hereinafter the displacement signal will be called indentation signal or simply indentation. In most cases it is necessary to make numerical fitting. These fittings require a high computational effort if the number of experimental measurements is very large. Hence the relevance of developing alternative processes such as the one proposed in the present invention. Figure 4 illustrates the experimental curve fitting to the PLR model assuming that the displacement signal is triangular. For an FDC curve taken at a speed v= 10 pm / s, the values Eo = 1616 Pa and y =0.14 are deduced (figure 4a). For an FDC curve taken at a speed v= 300 pm / s, the values o = 3237 Pa and y =0.16 are deduced (figure 4b). The theoretical curve of the PLR model reproduces the experimental curve at v =10 pm / s quite well. The precision of the fitting can be defined by the regression coefficient (R2=0.999). However, the theoretical fitting obtained on an experimental FDC curve of the same cell, but using a displacement signal with a speed of 300 pm / s (Figure 4b) is considerably worse (R2=0.971 ). The difference lies in the fact that at high speeds the profile of the displacement signal does not follow an exact triangular signal.

[0026] In the current state of the art, until now no machine learning methods have been developed that allow FDC curves to be processed, obtaining viscoelastic properties therefrom. This limitation prevents fast and simultaneous mapping of the topography and the nanomechanical properties of cells and tissues. There is, therefore, in the present technical field, the need to resolve this limitation, which would considerably increase the capabilities of force microscopes and their applications in biomedicine.

[0027] The present invention provides a solution to this need, through a novel method for processing curves generated with arbitrary displacement signals and obtaining viscoelastic nanomechanical properties in experiments carried out at different frequencies and speeds.

[0028] DESCRIPTION OF THE INVENTION

[0029] The present invention relates to a method for processing force versus distance curves obtained by a force microscope using a machine learning method based on functional multilayer perceptron neural networks. The method of the invention facilitates the prediction of mechanical and nanomechanical properties of soft materials, such as biological cells and tissues, that could be done in real time. In addition, it suppresses the errors associated with the nonlinearity of the displacement signal, which allows the precise calculation of several viscoelastic and nanomechanical properties, such as Young's modulus, compressive modulus or the fluidity coefficient.

[0030] It is an object of the present invention a machine learning method for quickly processing the FDC (force versus distance) curves obtained on viscoelastic materials and obtaining nanomechanical properties thereof. This object is achieved with the method of claim 1 . Different embodiments and aspects of the present invention are indicated in the dependent claims. More specifically, the present invention is based on combining two complementary elements. On the one hand, it consists of the use of nested neural networks for processing the FDC curves and obtaining the nanomechanical properties of the material. On the other hand, it considers linear (triangular) and non-linear indentation (sinusoidal, mixed, distorted) signals. Neural networks known as functional multilayer perceptron (functional MLP) are used as a machine learning method. In this network, the constituent elements are the layers, and within the layers, the neurons (nodes), in addition to the activation function. The first layer includes the input data fed to the network and the last layer are the results of the nanomechanical properties provided by the network. The intermediate layers (hidden layers) only have connections with the neurons of the first neighbor layers. The output of a neuron is a weighted sum of the neurons in the previous layer filtered by the activation function. The weighting coefficients are optimized to minimize variations with respect to the training data.

[0031] Figure 5 shows the general structure of the supervised machine learning system. It consists of two nested neural networks according to the invention. The first network is used to determine the fluidity coefficient and the second network provides the compressive modulus. Input parameters for the machine learning algorithm are the values of force, indentation, speed and thickness of the material.

[0032] The displacement or indentation signals considered here meet the following requirements: (1 ) The indentation is zero at the beginning and at the end of the process and takes always positive values. (2) The indentation is divided into two sections, approach and retraction. (3) The indentation has a maximum. (4) Indentation and time parameters are normalized to 1 . For this purpose t e (0, 1 ) and the maximum indentation is normalized to 1 .

[0033] More specifically, the method of the invention comprises carrying out the following steps: a) Generating hundreds of thousands of training FDC curves from the theoretical expressions of a viscoelastic model. The curves will be generated using different values of the model parameters, the thickness of the material, the value of the maximum indentation and the time in which the AFM tip is in contact with the material. b) Making forces and indentations dimensionless. Figure 6 shows several examples of the dimensionless theoretical curves used to train the network. The indentation signals for the above curves are shown in Figure 7; c) using indentation signals for training and validation, making time and indentation dimensionless, so their values remain in the (0,1 ) interval, allows to define all the indentation profiles used in AFM measurements, in accordance with the following mathematical conditions:

[0034] 1 ) The indentation reaches a maximum value at one point p of the normalized time interval (0, 1 ), i.e.,

[0035] I'(p) = 0 p e (0,1) (E.7) the indentation profile is also divided into approach and retraction sections where the derivative is, respectively, positive and negative:

[0036] (E. 8)

[0037] 2) The initial indentation takes a value very close to zero:

[0038] / (0) = e / (p) (E.9)

[0039] 3) The indentation takes a value of zero or very close to zero for t =1 :

[0040] 7(1) = e'Z(p) (E. 10) where e and e' are numbers close to 0; d) sampling the data from the training FDC curves. An example of the type of sampling used in the patent is provided in Figure 8; e) the aforementioned FDC curves will be used to train a machine learning method based on functional multilayer perceptron, wherein the first layer is formed by the input data and the last layer is formed by the output data. The intermediate layers allow non-linear interactions between neurons in neighboring layers to be carried out. The output of a neuron is a weighted sum of the values of the previous layer and it is filtered by an activation function; f) performing the predictions of the material parameters with two nested neural networks. The first neural network determines the fluidity coefficient. g) the above result is an input for the second neural network that is used to determine the compressive modulus; h) using mean absolute percentage error (MAPE) as a metric to quantify the accuracy of the method to predict the value of the compressive modulus

[0041] Wherein Etrue is the theoretical value and Epred is the value provided by the neural network. i) using the normalized root mean square error (NRMSE) as the metric to quantify the error of the fluidity coefficient wherein ytrue is the theoretical value and yPred the value provided by the neural network; j) completing the process when the above values have an error below 5%; k) applying the supervised machine learning method to experimental FDC curves obtained on the same material. This will require removing residual hydrodynamic contributions that are present in the experimental data; and l) obtaining the values of the compressive modulus and fluidity coefficient of the material from the experimental curves.

[0042] It is important to highlight the fact that the method of the invention, as stated in the previous lines, enables to process thousands of FDC curves in an automatic manner and very quickly.

[0043] In a preferred embodiment of the invention, the method comprises analyzing the experimental curves obtained on biological cells and tissues. In another preferred embodiment of the invention, in step (a) the theoretical results of the PLR model obtained for a spherical tip are used to train the machine learning method. In another preferred embodiment of the invention, the measurement is carried out having the sample immersed in a liquid, in a gas medium or in a vacuum.

[0044] Throughout the description and in the claims, the word “comprises” and its variants are not intended to exclude other technical features, additives, components or steps. For those skilled in the art, other objects, advantages and features of the invention may be deduced from both the description and the practical use of the invention. The following examples and drawings are provided by way of illustration and are not intended to limit the present invention. Furthermore, the present invention covers all possible combinations of particular and preferred embodiments included herein.

[0045] DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 shows a scanning electron microscopy image of a cantilever used in force microscopy to measure cells (model FastscanJD, Nano sensors).

[0047] Figure 2 shows an experimental example of a force versus distance curve obtained by AFM on a living HeLa cell. (a). The force dependence can also be represented with respect to time (b). The indentation or displacement signal with respect to the cell is shown in (c). The curves have been acquired at a speed of 10 pm / s. The approach and retraction sections are shown.

[0048] Figure 3 shows some examples of FDC curves generated theoretically by using the PLR model for a conical tip having an 18° semi-angle that approaches the cell at a speed of 10 pm / s and with different parameters as indicated on the panels, (a) Theoretical FDC curves for a material with Eo=1 OOO Pa and different fluidity coefficients (0.2, 0.4, 0.6). (b) Theoretical FDC curves for a material with y=0.2 and different compressive modulus (1000, 5000 and 10000 Pa).

[0049] Figure 4 shows the fitting of some experimental curves to the PLR model for a conical tip indenting the cell at different speeds, 10 pm / s (a) and 300 pm / s (b) if a triangular signal is assumed for the indentation. The experimental curves are represented by solid lines and the theoretical fittings by dashed lines. Figure 5 shows an example of the general structure of the nested neural networks used in the patent.

[0050] Figure 6 shows examples of the dimensionless theoretical force curves used to train the first neural network. Figure 6a shows the force as a function of time. Figure 6b shows the indentation and figure 6c shows the velocity.

[0051] Figure 7 provides an example of how theoretical curves are sampled to train and / or validate the machine learning method.

[0052] Figure 8 shows the detailed structure of the nested neural networks used in the patent.

[0053] Figure 9 shows six examples of the theoretical curves used for training and validation of the machine learning method.

[0054] Figure 10 shows the MAPE and NMRSE metrics when comparing a theoretical data set with the predictions obtained from the neural network (conical tip).

[0055] Figure 1 1 shows four experimental curves obtained on HeLa cells. Those curves have been extracted from a package of thousands of experimental curves (conical tip).

[0056] Figure 12 is an example of sampling carried out on an experimental force curve.

[0057] Figure 13 shows the MAPE (a) and NMRSE (b) metrics for FDC curves obtained experimentally by AFM in live HeLa cells.

[0058] Figure 14 shows MAPE and NMRSE metrics in a validation test using the neural network of the method of the invention (spherical tip).

[0059] Figure 15 shows MAPE and NMRSE metrics on predicting the modulus and fluidity coefficient on polyacrylamide gel (spherical tip).

[0060] PREFERRED EMBODIMENTS OF THE INVENTION

[0061] Reference numbers:

[0062] (1 ) Input data of the first neural network

[0063] (2) Hidden layers of the first neural network

[0064] (3) Input data of the second neural network

[0065] (4) Fluidity coefficient

[0066] (5) Intermediate layers of the second neural network

[0067] (6) Compressive modulus. Output data of the second neural network.

[0068] (7) Activation function

[0069] As described in the preceding sections, the present invention relates to a method based on the use of a supervised machine learning method for a fast and automatized method to process FDC curves obtained with an AFM on viscoelastic materials and predicting nanomechanical properties therefrom.

[0070] The process comprises three steps, training, validation and testing of the neural network to a set of experimental measurements. Preferably, the regressor is trained with 100,000 force-distance curves.

[0071] The training and validation of the neural network will be carried out on groups of theoretical curves determined by using the viscoelastic model known as PLR. To do this, a large number of curves obtained for different values of the model parameters, compressive modulus and fluidity coefficient, as well as different thicknesses of the material, are generated.

[0072] The processing of the curves involves the normalization of the variables to prevent the predictions from being affected by using curves of different force values, or by being generated by materials with different thickness. In particular, the force values of each curve used in training are normalized with the mean value of force in that curve, and the material thickness is normalized with respect to the maximum indentation used to generate that curve ( / max). This type of normalization will allow processing experimental curves with a very wide range of speeds (1 -1000 pm / s) and thicknesses (100 nm-100 pm). Examples of the theoretical curves are shown in figure 6.

[0073] By transforming all the variables into dimensionless variables, the neural network validity is not limited to a range of dimensional values.

[0074] The neural network determines the mean value of the dimensionless force (s), with s = t / ttot, being the dimensionless time, and ttotbeing the duration of the tip’s displacement (approach and retraction).

[0075] The nondimensionalization process can be expressed explicitly through the following transformations: with the modulus is given by wherein F and v are, respectively, the mean force and mean velocity of a FDC, and tr, Ir, are reference values with dimensions of time and distance, respectively. In addition, for a spherical tip |3o=3 / 2 and R is the sphere radius while for a conical [3o=2 and R is replaced by tan20.

[0076] The force values are sampled at constant time intervals (Figure 7). While the displacement or indentation are determined by the interpolation coefficients associated with the base of functions known as splines.

[0077] The intrinsic viscoelasticity of the material makes the force measured in experiments a functional of the indentation history. In other words, the force at any given moment depends on the previous deformation history. For this reason, the method of the invention uses functional data analysis, and in particular functional neural networks to predict mechanical properties from force-distance curves.

[0078] The incorporation of functional data analysis into a neural network requires the implementation of a functional neuron. The functional neuron replaces the discrete weights found in common neurons with a weight function.

[0079] The scalar product of the functional input / -(x) and the weight function #(x) is computed as: Basis expansion is used to facilitate backpropagation and compute the gradient of the loss function with respect to the weight functions:

[0080] The functional inputs can also be expanded in the same basis:

[0081] Then, Eq. (a) can be solved by computing a Gram matrix associated to the basis functions, the scalar product is equivalent to the discrete counterpart with the metric given by Gtj,

[0082] Treating the expansion coefficients cl)as scalar inputs gives more freedom for training. The Gram matrix can be absorbed into the network weights and the coefficients can be standardized to facilitate the training of the regressor. However, this implementation requires a pre-processing step of spline interpolation.

[0083] Using this representation, we can incorporate the indentation and the velocity as functional inputs for the network.

[0084] A different scheme is applied to the force. For each force curve, the force values are sampled to a fixed size using linear interpolation and non-dimensionalized to have mean value of 1 . The sampling size is optimized as a hyperparameter. In contrast to the expansion coefficient, the force is normalized using group normalization.

[0085] Thus, the force is uniformly sampled with (respect to time) and the indentation and the velocity are expressed through a basis expansion.

[0086] The predictions of the fluidity coefficient given by the outer network together with the indentation, the velocity and the cell thickness are the inputs of the second neural network.

[0087] Different types of deformation profiles might be used in AFM to obtain a forcedistance curve. That feature should be considered to train the machine learning method. For this purpose, a sampling scheme is used to generate indentation profiles of arbitrary shapes, then the corresponding force is computed by using the theoretical model previously explained. where N+ 1 = m + 4 is the number of basis functions and m is the number of knots of the splines. This particular choice of basis functions allows to translate the restrictions in the functional space to a system of linear equations and inequalities that define a polytope in Euclidean space. Finally, this volume is sampled uniformly generating an artificial set of indentation profiles for training.

[0088] To generate the data for training, the indentation profiles are represented using basis expansion. It has been chosen b-splines of degree 3 as the basis functions to generate the indentation profiles. Splines are piecewise polynomials joined in a set of points, called knots, by imposing continuity and differentiability. The degree of the polynomials determines how many times the spline can be differentiated. This basis of b-splines is defined by the following expressions:

[0089] Furthermore, these functions meet the following property if f —x— f+k+1

[0090] (E. 19) otherwise where ttdenotes the knots of the splines, and k denotes the degree of the spline. In other words, they have compact support and are positive over the whole domain. In this case the number of nodes / is 15 and the degree of the polynomials k is 3. From the above, the indentation is developed on the base of splines. A degree of 3 and 40 equidistant interior knots is used in the interval (0,1).

[0091] To simplify the calculations, it is considered that E= 0, which implies

[0092] The rest of the conditions are easier to be applied over the derivative basis. The velocity of indentation is also expressed on the base of splines as:

[0093] Wherein Mij is a square matrix (N x N) which relates the subspace of derivatives to the base of splines having degree 2. In particular, the inequalities presented in equation (E.8) can be met using the fact that the basis of splines is positive throughout its domain:

[0094] (E. 22)

[0095] Wherein / * is an index based on the position of the maximum (p) that takes into account the compact support of the function base. Although this solution is more restrictive than equation (E.8), it covers all experimental cases since the functions outside this solution are extreme cases.

[0096] With the foregoing, equation E.7 takes the form of equation 23:

[0097] = 0 (E. 23) and equation 10 takes the form of equation 24:

[0098] Finally, the normalization condition / (p) = 1 is enforced. In principle, the absolute value of the coefficients di is not limited. By enforcing this normalization we can reduce this infinite space to a finite volume, such as / di / belongs to the interval (0,1 ) , which is equivalent to the entire space under the normalization by dividing the coefficients by / (p).

[0099] The set of equations E.20, E.22, E.23 and E.24 form a set of linear equalities and inequalities in the coefficients {c0, however, in principle the coefficients d, are not bounded. The key point for reducing the space of possible solutions to a finite volume is the fact that the restrictions imposed are scale invariant. This property implies that the sets defined as Sd. = {a dt\a e IR} form equivalence classes. This property makes it possible to sample a finite volume such as |d e (0,1) and collapse all the solutions to a unique representative of the class by imposing the normalization:

[0100] Note that the volume selects forms a polytope in IRWonly for fixed values of p and e'. This means that p and e' are additional degrees of freedom which have to be specified. In particular, e' can be chosen as zero without losing generality, and p = 0.5 for all the operational modes used to obtain force-distance curves in AFM. However, in order to simulate noise and instrument limitations in the experimental data e' was sampled in the interval (-0.005, 0.005) and p was sampled in the interval (0.4, 0.6). Finally, the volume defined by the system of equations described above is uniformly sampled using a form of rejection sampling.

[0101] Summing up, the sampling of the indentation can be broken down into the following steps:

[0102] 1 . Selecting a value of p and in the range of experimental values. Defining the value of the corresponding index / *.

[0103] 2. Uniformly sampling all the available degrees of freedom, \di\ values in the interval (0,1 ) and assigning the corresponding sign according to equation 22.

[0104] 3. Solving the System of equations E.23 and E.24. Verifying that inequalities E.22 hold and that the solution is inside the sampling volume verifies \di\ < 1 . Otherwise, repeating steps 2 and 3 until a solution is found that meets the same.

[0105] 4. Defining the coefficients M by calculating the inverse matrix-1and fixing co = 0.

[0106] 5. To complete the normalization process, all the previous a and di coefficients are divided

[0107] The curves are generated by applying a random sampling of values for y, and the dimensionless cell thickness (rmaxI h) in the interval (0, 1 ) where, for a spherical tip, rmax= J R ■ lmaxis the maximum radius of contact.

[0108] The aforementioned coefficients allow the indentation and velocity profiles to be represented in a functional manner to be used as inputs of the neural networks. A detailed description of the inputs, intermediate steps and outputs of the nested neural network system is shown in Figure 8.

[0109] In a preferred embodiment of the invention, the nested neural network system of Figure 8 is trained with theoretical curves such as those shown in Figure 9. In a preferred embodiment of the invention the reference values are established as tr= tmax, Ir = Imax and vr=v, (average velocity). Figure 10a compares the results obtained by the neural network system for the compressive modulus obtained with that of the theoretical curve values. An excellent agreement is observed between them, quantified by a MAPE= 0.3%. Moreover, figure 10b shows the comparison between the theoretical values and those predicted by the neural network system for the fluidity coefficient, NRMSE= 1.3%. With the neural network system trained and validated, it is tested using input values from experimental AFM curves.

[0110] In a preferred embodiment of the invention, a set of experimental FDC curves obtained with an AFM on HeLa cells are delivered to the nested neuronal networks. Figure 1 1 shows four experimental curves extracted from a package of thousands of curves on which the nanomechanical properties thereof are to be predicted. Figure 12 provides an example of how the experimental curves are sampled.

[0111] Figure 13a compares the results obtained by the neural network system for the compressive modulus with that of the values obtained by a model fitting. A good agreement is observed between them, quantified by a MAPE= 3.4%. Moreover, figure 13b shows the comparison between the numerical values and those predicted by the neural network system for the fluidity coefficient, NRMSE= 4.6%.

[0112] A grid search with 5-fold cross validation is performed to select the network.

[0113] Figure 14 shows MAPE and NMRSE metrics in a validation test using the neural network of the method of the invention wherein the parameters Eo and y are predicted with an accuracy of about 1 %. In figure 14, the dashed line indicates the true values and the points the predicted values. Figure 15 shows MAPE and NMRSE metrics on predicting the modulus and fluidity coefficient on polyacrylamide gel comparing with AFM experimental data obtained with a spherical tip R= 3.31 pm. In this case, EQ is predicted with an accuracy below 6% and y with an accuracy below 1 %.

Claims

CLAIMS1 A computer-implemented supervised machine learning method for processing force versus distance curves obtained by AFM on viscoelastic materials, wherein the method comprises the steps of: a) obtaining experimental force versus distance curves within a viscoelastic material; b) generating a number of training force versus distance curves from the theoretical expressions of a model of the viscoelastic material; c) determining the parameters of the viscoelastic material using functional neural networks wherein: c.1 ) a first neural network is configured to predict the fluidity coefficient of the viscoelastic material; and c.2) a second neural network is configured to predict the compressive modulus of the viscoelastic material; such that the values of the fluidity coefficient of the material obtained in the first neural network are established as input values of the second neural network to predict the compressive modulus; d) quantifying the precision obtained in the prediction of the fluidity coefficient and the compressive modulus of the viscoelastic material of step (c); e) completing the training process when the value provided by the first and second neural networks reaches an error or deviation from the theoretical value of the fluidity coefficient and the compressive modulus of less than 5%; and f) applying the machine learning method from step (e) on the experimental force versus distance curves, obtained within the same viscoelastic material, to determine the values of the compressive modulus and fluidity coefficient of the viscoelastic material from the experimental curves.2.- The machine learning method according to claim 1 wherein the training curves are generated from a theoretical viscoelastic model3.- The machine learning method according to claim 1 comprising: using a metric of the averaged relative error: mean absolute percentage error (MAPE), to quantify the error of the method to predict the value of the compressive modulus; and using the normalised standard deviation to quantify the error obtained to predict the fluidity coefficient.4.- The machine learning method according to any of the previous claims, wherein displacement or indentation signals are considered those signals that meet: (a) The signal is zero (or very close) at the beginning and at the end of the indentation and takes always positive values; (b) the signal is divided into two sections, approach and retraction, (c) the indentation has a maximum; (d) spatial and time signals are normalized to 1 , wherein t belong to the interval (0,1 ) and the maximum indentation is normalized to 1 .5.- The machine learning method according to claim 4 comprising a step of normalization of the variables of each force versus distance curve, wherein: the force values are normalized with the average value that the force takes in the approach and retraction sections; the time is normalized to the value of the time tr at which the maximum indentation / = Imax, is reached, the indentation is normalized to its maximum value lr = Imax the velocity is normalized to its average value v6.- The method according to any of the preceding claims, wherein the parameters of the experimental force versus distance curves are extracted by using a linear viscoelastic model.7.- The method according to any of the preceding claims, wherein a viscoelasticmodel described by a relaxation function of the following type is used:wherein r is the Euler gamma function; fo is a reference time, as an example fo = 1 s is taken and EQ is the compressive modulus; y is the fluidity coefficient.8.- The method according to any of the preceding claims, wherein the parameters of the experimental force versus distance curves are extracted by using a viscoelastic model that includes bottom-effect corrections.9.- The method according to any of the preceding claims, wherein the parameters of the experimental force versus distance curves are extracted from force versus distance curves that include the calculation of the contact area during the retraction of the section by using the Tings’ model.10.- The method according to any of the preceding claims, wherein the measurement is carried out having the sample immersed in a liquid, in a gas medium or in a vacuum.1 1 .- The method according to any of the preceding claims wherein the viscoelastic material are biological cells or tissues.12.- The method according to any of the preceding claims wherein the viscoelastic material is a polymer.13.- The method according to any of the preceding claims wherein the training curves are generated by expressing indentation as a functional input for the neural networks to which a scalar product of a weight function of the neural networks is applied in the form:and wherein basis expansion and Gram matrix are used for discretizing said expression resulting:wherein the expansion coefficients cl)are scalar inputs sampled in the training of the neural networks.14.- The method according to claim 13, wherein the number of basis functions used is N + 1 = m + 4, wherein m is the number of knots of the splines.15.- The method according to any of the preceding claims wherein the training of the neural networks is performed by sampling the force values to a fixed size optimized as hyperparameter.16.- The method according to any of claims 4 to 15, wherein the constraints imposed collapse all solutions of indentation profiles to a unique representative class forming a polytope in IRWonly for fixed values of p and e', wherein e' is an error in the indentation at t=1 and is sampled in the interval (-0.005, 0.005) and p is the value of t in the maximum indentation value and is sampled in the interval (0.4, 0.6).17.- The method according to claim 16, wherein the indentation is uniformly sampled using a form of rejection sampling.18.- The method according to any of the preceding claims wherein a grid search with 5-fold cross validation is performed to select the network.

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