Optimal design for multi-compartment cell elasticity estimation using AFM and super-resolution imaging

Abstract

Background and Objective.

The transduction of external forces to the internal components of the cell has critical implications for cell behavior in health and disease. Despite advances in techniques such as atomic force microscopy (AFM), accurately and reproducibly estimating the mechanical properties of multiple individual subcellular compartments remains challenging. This is often due to variability in measurements, the lack of a standard “inverse” approach to estimate unknown properties, and an often ill-posed inverse problem. This study presents an integrated experimental–computational framework for the optimal design of an inverse approach to estimate multi-compartment cell properties, focusing on the behavior of the nuclear membrane, cytoplasm, and nucleoplasm. The optimal design approach identifies an ideal set of AFM measurements that minimizes the dependence of the estimated multi-compartment properties on AFM probing locations, thereby improving the reproducibility of mechanical property estimations.

Methods:

Super-resolution imaging was used to construct a 3-D computational model of a human umbilical vein endothelial cell (HUVEC). An inverse modeling approach was then used to fit experimental data to a hyperelastic constitutive model, accounting for large-deformation nonlinearities. Simulations incorporating visco-hyperelasticity were conducted to explore viscous effects.

Results:

Our approach leads to quantification of the mechanical properties of the nucleoplasm, nuclear membrane, and cytoplasm with minimal dependence on loading conditions.

Conclusions:

We expect our approach to assist with standardizing the biomechanical characterizations of subcellular structures, improve the consistency and reproducibility of the estimations across mechanobiological studies, and ultimately improve our understanding of the role of mechanotransduction in disease progression.

1. Introduction

Cell mechanobiology examines how mechanical forces shape cellular behavior and function, revealing the interplay between physical cues and biological responses [1,2]. Cells exist in a complex physical microenvironment and are continually subjected to varying mechanical forces and deformations. The transmission of such stimuli to mechanosensitive cell components plays a fundamental role in orchestrating various cell processes, including proliferation, differentiation, migration, and tissue homeostasis [3–6]. The cell nucleus serves as a central mechanosensitive cellular structure that relies on molecular transducers to detect external signals and adjust intra-nuclear chromatin organization, which, in turn, governs cell function [7]. Recent studies have revealed that alterations in intrinsic cellular mechanical properties can critically affect cellular sensing and stress transduction [8,9]. As the transmission of physical signals to the nucleus has been shown to be a notable driver of cell adaptation in disease [9–12], accurate characterization of longitudinal changes in the mechanical properties of cellular components, and their effect on transduction, is crucial in furthering our understanding of disease progression and could elucidate new therapeutic targets. Accurate and reproducible measurement of multi-compartment cell properties has proven challenging; however, we hypothesize that an optimal design of experiments (ODE) approach could provide a means to minimize dependence on specific loading conditions and consistently estimate the mechanical properties of structural cellular compartments.

Traditional experimental approaches to characterize the mechanical properties of cells often face challenges due to the delicate nature of cellular structure and the limitations of available techniques [13]. Atomic force microscopy (AFM) is a widely used method for characterizing the mechanical behavior of cell components, given its ability to apply controlled forces, high sensitivity, and ease of sample preparation [3]. AFM involves characterizing the force-indentation relationship using a flexible cantilever, which acts as a spring, applied to the cell. Cell elasticity is often described as the apparent Young’s or elastic modulus, which is obtained by fitting the force-indentation curve. The most common methods for measuring nuclear stiffness involve probing the local stiffness above the nuclear region of an intact cell [14,15]. More advanced techniques, such as AFM force mapping, provide a wealth of spatial data by collecting force curves over a grid [16–18], a method that has enabled the estimation of properties for distinct cellular sub-components, including nucleoli [19,20]. However, these measurements, whether from a single point or a map, can be biased by the presence of subcellular structures, such as the plasma membrane and cytosolic content, between the nucleus and the AFM probe. Alternative techniques, such as isolating the nucleus [21,22] or using a needle-tip AFM probe to penetrate the plasma membrane [23,24], can provide more direct measurements. Yet removing the nucleus from its natural cellular context may lead to deviations in actual nuclear behavior. While methods for measuring cell compartment elasticity via AFM have improved significantly since its invention, studies indicate that measurements of individual cell components can vary drastically depending on probe shape and measurement location [25,26]. Moreover, the absence of a standardized approach to multi-compartment biomechanical characterization has led to inconsistencies in reported cell properties [25,27]. In-silico finite-element (FE) modeling that accounts for real 3-D cell geometry offers a promising avenue to overcome these obstacles. In particular, leveraging computational simulations and optimization algorithms can extend established experimental methods to differentiate and estimate the mechanical properties of critically understudied cellular compartments, such as the nuclear membrane, nucleoplasm, and cytoplasm, in situ with unprecedented precision.

Elasticity determined via AFM characterizes a global modulus, given that the captured force-indentation relation describes the overall integrated stiffness of the cellular substructures within the cell membrane. AFM has been extensively employed to characterize the global elasticity of both the cytoplasm and nuclear compartments of cells [8,9,28]. These studies have emphasized the crucial role of cellular biomechanical sensing mechanisms in regulating cell behavior in aging and cancer, among other diseases [7]. Such findings suggest that analyzing the mechanical properties of cell substructures beyond the commonly studied components, such as the cytoplasm and nucleus, could uncover new interactions between cellular biomechanical remodeling, sensing, and behavior. However, a reliable characterization of these structures by solving the underlying inverse problem is challenging, as it requires multiple AFM measurements at different locations on a single cell to reduce parameter uncertainty brought about by specific loading conditions and complex compartment geometries. Even so, the simple addition of further measurement locations may introduce redundancy and potential parameter correlation, which can obscure the identification of distinct mechanical properties across compartments, leaving the problem ill-posed due to non-unique solutions. As such, a well-posed solution depends not on the quantity of data but on acquiring informative data that can de-correlate parameter codependence. Thus, there is a need for a standardized method of designing AFM procedures to ensure a robust estimation of the parameters while minimizing potential interdependence of parameters. An optimal characterization approach leveraging computational modeling holds promise to inform experimental procedures and enable the most reliable biomechanical characterization of sub-cellular compartments.

In this work, we have developed an integrated experimental-computational approach to perform the optimal characterization of cell multi-compartment elasticity, consisting of cytoplasm, nucleus, and nuclear membrane, using a minimal number of probing locations. We directly confront the challenge that estimating multiple parameters in such complex, hierarchical models is often an ill-posed inverse problem, prone to non-unique solutions due to parameter codependence. We make use of super-resolution imaging of human umbilical vein endothelial cells (HUVECs) to develop a 3-D computational model of the cell, delineating the three separate compartments. We assign a classical Fung-type exponential constitutive model to individual compartments, enabling us to capture the strong nonlinearity commonly observed in the cell mechanical behavior at finite strains, moving beyond the prevailing paradigm of characterizing cell stiffness via apparent Young’s modulus. We use the D-optimality criterion to identify the optimal AFM probing locations. This framework provides a rigorous method to ensure the parameter estimation problem is optimal by finding the experimental design that minimizes parameter correlation and maximizes their unique identifiability. We then use an inverse problem approach to estimate and report constitutive parameters characterizing the hyperelastic mechanical behavior of the cellular compartments. We show that the parameters identified by non-optimal or poorly-conditioned AFM-modeling technique are not unique and cannot predict the cell behavior under new loading conditions. On the other hand, our proposed computational approach leverages in-silico modeling and identifies the ideal set of AFM data that results in estimated parameters that can reliably predict the cell behavior under general conditions. Our approach involves three steps: i) in-silico inverse modeling with AFM data at several sites on the image-based cell model, ii) assessment of D-optimality to identify the optimal number and location of probing sites and estimation of the parameters using an inverse problem approach, and (iii) validating the superior predictive capability of the optimal set over a non-optimal set of parameters. The presented approach will facilitate accurate, reproducible characterization of previously understudied cellular compartments in health and disease, ultimately elucidating new mechanisms underlying mechanosensitive cellular diseases. It provides a formal method to manage the complexity and potential data redundancy of advanced techniques such as force mapping, ensuring that the chosen experimental design, whether it consists of a few points or a full map, is optimized for a well-conditioned inverse problem and robust parameter identification.

2. Materials and methods

An important question arising in AFM procedures is the selection of probing sites to measure the force-indentation response (Fig. 1), particularly because studies have shown that measurements can vary significantly depending on the chosen measurement location [25,26]. To aptly address this question, a methodology is required involving (i) collection of experimental data via AFM at multiple sites, (ii) identification of the “best” AFM data, which will lead to robust parameter estimation, and (iii) implementation of an integrated inverse problem to infer reliable estimates for the constitutive parameters. In the following sections, we demonstrate the need for optimal parameter estimation to estimate cell elasticity and present a methodology to address the uncertainties introduced by the prevailing AFM techniques.

Fig. 1.

Robust and reproducible characterization of the biomechanical response of cellular compartments from an intact cell requires multiple atomic force microscopy (AFM) measurements, and the estimated elasticity depends critically on the design of AFM experiments. Our integrated experimental–computational approach leverages in-silico modeling to identify “optimal” AFM data for estimating parameters that accurately characterize biomechanics under general loading conditions. Created in BioRender.

We used HUVECs for our AFM studies because they are readily obtained and cultured. Endothelial cells are exposed to and respond to hemodynamic forces (e.g., oscillatory shear stress and cyclic strain) that are physiologically significant. Disorders in the response of endothelial cells to these forces contribute to cardiovascular disease [29,30]. Consequently, our AFM studies of HUVECs may have both physiological and pathophysiological significance. Furthermore, there is a rich literature describing the acute and chronic effects of physical forces on endothelial cell cytoplasmic signaling and gene expression [31–36].

We first collected AFM force-indentation measurements at two locations of a HUVEC: one above the nucleus (nuclear location) and one far from the nucleus (cytoplasmic location). An in-silico 3-D two-compartment cell model was developed. Next, an inverse modeling approach, accounting for the nonlinear mechanical properties, was used to derive constitutive parameters for the nucleus and cytoplasm from the experimental data. We show that the parameters identified by this established AFM-modeling technique are not a set of robust, unique parameters; thus, an improved methodology is needed to infer reliable parameter estimates. To address this, we present an approach for determining the optimal AFM probing sites. A similar optimal design approach has been used previously to characterize the 3D passive kinematics of normal myocardium and myocardium with hydrogel inclusions [37–39]. Additionally, we extend the inverse problem and optimal estimation approach to a three-compartment cell model (cytoplasm, nucleus, and nuclear membrane) to demonstrate the potential of this technique to improve the reliability of biomechanical characterization of additional cellular substructures.

2.1. Experimental procedures

Cell harvesting and preparation

Isogenic HUVECs were obtained and cultured as per previously published [9]. Cells were seeded onto glass cover-slips. For reliable cell attachment, cover-slips were $O_2$ plasma treated and incubated with an extracellular matrix (ECM) solution of rat tail collagen type I and human fibronectin.

Atomic force microscopy

We used AFM data previously acquired by our colleagues Walther et al. [9]. AFM measurements were performed using a Catalyst Biomicroscope (Bruker) with a heating apparatus to maintain cell cultures at 37 °C. A spherical probe (SiO 2 bead, silicon cantilever, radius 1.25 μm, spring constant 0.05 N/m, NovaScan Technologies) was used for all live-cell mechanical measurements. Two measurements were taken per cell, at a nuclear and a cytoplasmic location, with several replicates performed sequentially at each site. To reduce the influence of stiff bottom effects, force-indentation curves up to ~ 20% of the maximum force were used for parameter identification, corresponding to an indentation depth of 0.710 μm and 0.243 μm at the nuclear and cytoplasmic probing locations, respectively. Measurements were made with an indentation speed of 20 μm/s over a duration of 0.5 s. A complete list of all AFM indentation parameters is included in the supplement (Table S1). Further details regarding the AFM and probe location procedures are described in [9].

Cell labeling

After completing AFM measurements and topographic imaging, cells were washed and stained for subsequent super-resolution imaging. Cells were washed with PBS and fixed with 4% paraformaldehyde. Samples were then blocked and permeabilized with a blocking buffer. Cells were then stained in a two-step process: (i) samples were incubated with mouse anti-Lamin-A (Invitrogen) and rabbit anti-paxillin (phospho-Y31, Abcam); (ii) secondary staining was performed with donkey anti-rabbit Alexa Fluor 555 (Invitrogen Molecular Probes, Thermo Fisher Scientific) and donkey anti-mouse Alexa Fluor 647 (Invitrogen Molecular Probes, Thermo Fisher Scientific). Finally, actin and DNA were stained with phalloidin Alexa Fluor 488 (Invitrogen Molecular Probes, Thermo Fisher Scientific) and 4’,6’-diamidine-2’-phenylindole dihydrochloride (DAPI, Roche Diagnostics), respectively. Further details regarding the cell staining procedures are described by Walther et al. [9]

Super-resolution imaging

Super-resolution imaging was performed using a Nikon n-SIM Structured Illumination Super-Resolution Microscope System equipped with a Hamamatsu ORCA-Flash, v.4.0, V2 digital camera (Nikon Instruments). Bright-field correlative imaging was performed using a CFI Plan APO Lambda 20x objective lens (Numerical aperture: 0.75, Nikon). Super-resolution z-stack imaging was performed using a CFI SR APO total internal reflection fluorescence (TIRF) AC 100XH oil-immersion objective lens. Z-stack spacing was set to 100 nm. Images were reconstructed by analyzing a documented moiré pattern in NIS Elements (Nikon) v. AR 5.30.03.

The use of super-resolution structured illumination microscopy (SIM) for reconstruction of the HUVEC geometries provided high-resolution 3D representations of cell geometries for subsequent FE modeling. Indeed, the quality of the 3D reconstruction is crucial to our modeling approach, and SIM may not be optimal for all cell types, such as those with poorly defined cytoskeletal structures. For this reason, our inverse FE approach is agnostic to the source of the 3D geometry. This framework is designed to be adaptable and can readily incorporate 3D cell models generated by any high-resolution imaging technique, including STED (Stimulated Emission Depletion), localization-based microscopy (e.g., PALM, STORM), and even high-resolution confocal microscopy.

2.2. 3-D finite element cell models

Two- and three-compartment cell FE models were used to numerically simulate the behavior of the cell in response to AFM force indentation at different probing locations. FE simulations were conducted using ABAQUS. The key steps for creating these FE models and performing inverse estimation are detailed below.

Image processing and geometry reconstruction

Super-resolution image stacks for the nucleus and cytoplasm were contoured using LabelMe [40] with 100 nm between each Z slice. 3-D geometries of the cell compartments (nucleus and cytoplasm) were reconstructed from the contours and then finely meshed with linear tetrahedral elements (Fig. 2). In the extended three-compartment model, the nuclear membrane was modeled using triangular shell elements embedded at the interface between the cytoplasm and nuclear compartments with a prescribed thickness of 35 nm [41,42]. This additional component of the model represents the combined effect of the lipid membrane and the nuclear lamina. The AFM probe was modeled as a thick-walled rigid sphere with an outer radius of 1.25 μm and a thickness of 0.15 μm.

Fig. 2.

(A) Meshed 3-D geometry of the human umbilical vein endothelial cell (HUVEC) with the AFM spherical probe showed at the two probing locations used in the experimental setup. (B) Visualization of the cytoplasm and nuclear compartment (nucleoplasm and nuclear membrane) of the HUVEC in the two-compartment model. (C) Visualization of the cytoplasm, nuclear membrane, and nucleoplasm of the HUVEC in the three-compartment model.

Boundary conditions

The bottom cell surface was constrained from movement in all directions, representing a rigid, non-deformable plane at the base of the cell geometry. This boundary condition simulated the glass substrate used in the AFM experiments and accounted for potential stiff bottom effects. The rigid beads were located at sites just above the cell surface. A general contact interaction using the linear penalty method was used to simulate the contact between the rigid bead and the cell surface. The beads were then prescribed a displacement along the indentation axis equal to the displacement applied in the AFM experiments, and reaction forces were calculated directly from the node at the center of the rigid bead.

Visco-hyperelastic constitutive modeling

Viscoelastic FE simulations were performed to investigate how time- and rate-dependent mechanical effects influence the optimal design approach. To capture the viscoelastic behavior of the nucleus and cytoplasm, we utilized a standard linear solid (SLS) model [43] formulated through two thermodynamic potentials denoted by $\psi$ and $\varphi$. The free energy function:

$$\psi \left(\mathbf{F},{\mathbf{C}}^{\nu }\right)={\psi }^{Eq}\left(\mathbf{F}\right)+{\psi }^{NEq}\left(\mathbf{F},{\mathbf{C}}^{\nu }\right),$$ (1)

describes the energy stored via elastic deformation. In this expression, ${\psi }^{Eq}$ is a non-negative function characterizing elastic energy storage at thermodynamic equilibrium, while ${\psi }^{NEq}$ represents the additional energy storage at non-equilibrium states (i.e., the portion eventually dissipated). The symmetric second-order tensor ${\mathbf{C}}^{\nu }={\mathbf{F}}^{\nu T}{\mathbf{F}}^{\nu }$ serves as an internal state variable representing the viscous component of the deformation gradient $\mathbf{F}$ [43]. Energy dissipation through viscous deformation is governed by a dissipation potential of the form:

$$\varphi \left(\mathbf{F},{\mathbf{C}}^{\nu },{\dot{\mathbf{C}}}^{\nu }\right)=\left\{\begin{array}{ll}\frac{1}{2}{\dot{\mathbf{C}}}^{\nu }\cdot 𝒜\left(\mathbf{F},{\mathbf{C}}^{\nu }\right){\dot{\mathbf{C}}}^{\nu }& \text{if det}{\mathbf{C}}^{\nu }=1\\ +\infty & \text{otherwise}\end{array}\right.,$$ (2)

where ${\dot{\mathbf{C}}}^{\nu }$ is the material time derivative of ${\mathbf{C}}^{\nu },𝒜$ is the fourth-order viscosity tensor, and the dot operator denotes double contraction. This viscosity tensor characterizes the deviatoric viscosity and is defined as:

$${𝒜}_{\mathit{\text{ijkl}}}\left(\mathbf{F},{\mathbf{C}}^v\right)=\frac{\eta }{2}{\left(C_{il}^v\right)}^{-1}{\left(C_{jk}^v\right)}^{-1},$$ (3)

where $\eta$ is a non-negative scalar material viscosity function. The rheological representation of the SLS model and the corresponding mechanical parameters are illustrated in Fig. 3.

Fig. 3.

Standard linear solid (SLS) model implemented for viscoelastic simulations. The schematic illustrates the material parameters $\left\{a_1,a_2,b_1,b_2\right\}$ that govern the hyperelastic behavior of the SLS model via the strain energy function defined in Eq. (4). The parameters $a_C,b_C,a_N$, and $b_N$ were obtained through the inverse estimation approach described in Section 2.2. The SLS parameters $\left\{a_1,a_2,b_1,b_2\right\}$ were calculated such that, at the limit of negligible viscosity, the equilibrium and non-equilibrium contributions (Eq. (1)) equal the total hyperelastic behavior of the compartment defined by $\left\{a_C,b_C,a_N,b_N\right\}$.

In our implementation, ${\psi }^{Eq}$ and ${\psi }^{NEq}$ represent the passive mechanical behavior of the cellular compartments that were modeled as isotropic, hyperelastic, nearly incompressible materials. We used a classical exponential Fung-type constitutive law [44] of the form:

$$\begin{gathered} {\psi }_i^{Eq}(\mathbf{F})=\frac{a_i}{2b_i}\left[e^{b_i\left({\bar{I}}_1-3\right)}-1\right]+\frac{\kappa }{2}(J-1);i=C,M,N, \\ {\psi }_i^{NEq}\left(\mathbf{F},{\mathbf{C}}^{\nu }\right)=\frac{a_i}{2b_i}\left[e^{b_i\left({\bar{I}}_1^e-3\right)}-1\right];i=C,M,N, \end{gathered}$$ (4)

where $a_C,a_M$, and $a_N$ are positive material constants governing the contribution of the cytoplasm $\left(C\right)$, nuclear membrane $\left(M\right)$, and nucleus $\left(N\right)$, respectively. Here, $J=\text{det}\left(\mathbf{F}\right)$ denotes the volumetric changes of deformation, and $\kappa$ was chosen sufficiently large to enforce incompressibility. In the two-compartment model, the nuclear parameter $a_N$ includes the behavior of both the nuclear membrane and nucleoplasm, whereas in the three-compartment model, the nuclear parameter $a_N$ governs only the nucleoplasm, while the membrane parameter $a_M$ governs the nuclear membrane. The kinematic variable ${\bar{I}}_1$ is defined as ${\bar{I}}_1=\text{tr}\stackrel{-}{\mathbf{\text{C}}}$, where $\mathbf{C}$ is the right Green deformation tensor, and $\overline{\mathbf{C}}=J^{-2/3}\mathbf{C}$. The variable ${\bar{I}}_1^e$ is defined as ${\bar{I}}_1^e=J^{-2/3}\text{tr}\left[\mathbf{C}{\left({\mathbf{C}}^{\nu }\right)}^{-1}\right]$. The total Cauchy stress $\mathbf{T}$ was calculated as:

$$\mathbf{T}=J^{-1}{\mathbf{S}\mathbf{F}}^T,$$ (5)

Here, $\mathbf{S}$ is the first Piola–Kirchhoff stress tensor at any material point $\mathbf{X}$ and time $t\in [0,T]$ given by:

$$\mathbf{S}(\mathbf{X},t)=\frac{\partial \psi }{\partial \mathbf{F}}\left(\mathbf{F},{\mathbf{C}}^{\nu }\right),$$ (6)

where ${\mathbf{C}}^v$ is defined implicitly by the evolution equation:

$$\left\{\begin{array}{l}\frac{\partial \psi }{\partial {\mathbf{C}}^{\nu }}\left(\mathbf{F},{\mathbf{C}}^{\nu }\right)+\frac{\partial \varphi }{\partial {\dot{\mathbf{C}}}^{\nu }}\left(\mathbf{F},{\mathbf{C}}^{\nu },{\dot{\mathbf{C}}}^{\nu }\right)=0\\ {\mathbf{C}}^{\nu }(\mathbf{X},t=0)=\mathbf{I}\end{array}\right..$$ (7)

A detailed mathematical implementation of this framework is provided by Lefèvre et al. [43]. The parameter set implemented in the viscoelastic simulations $\left\{a_1,a_2,b_1,b_2\right\}$ corresponds to the material parameter values that are attributed to each of the hyperelastic “spring” components of the SLS model (Fig. 3). Our approach initially estimates the parameters $a_C,b_C,a_N$, and $b_N$, neglecting any viscous behavior (as described in the following section). When applying these estimated parameters in the viscoelastic analysis, $\left\{a_1,a_2,b_1,b_2\right\}$ are calculated via the relations defined in Fig. 3 to ensure that, at the limit of negligible viscosity, the equilibrium and non-equilibrium contributions (Eq. (1)) equal the total hyperelastic behavior of the compartment defined by $\left\{a_C,b_C,a_N,b_N\right\}$.

Inverse estimation of material parameters

An iterative least-squares optimization routine was used to estimate the material parameters that provide the most physically accurate representation of the behavior of cell compartments [45]. Only hyperelastic constitutive parameters were estimated (viscous behavior was neglected for simplicity). The forward problem involved indenting the cell surface with rigid beads. The inverse problem was initiated by prescribing an initial guess for the parameter set $\mathit{\gamma }$. The optimal parameter set ${\mathit{\gamma }}^{*}$ was obtained by minimizing the sum of squared differences between the simulated and experimental force-indentation responses:

$${\mathit{\gamma }}^{*}=\text{arg}\underset{\mathit{\gamma }}{\text{min}} {‖F_{\text{Sim}}\left(\mathit{\gamma }\right)-F_{\text{Exp}}‖}^2,$$ (8)

where $F_{\text{Sim}}\left(\mathit{\gamma }\right)$ is the simulated force-indentation curve given a set of material parameters $\mathit{\gamma }$, and $F_{\text{Exp}}$ is the corresponding experimental measurement. To resolve the inherent parameter codependence between $a$ and $b$ in the constitutive model [46,47] and ensure a unique solution, we fixed $b=1.5$ for all compartments based on pilot simulations. Inverse estimation approaches for the two- and three-compartment models proceeded as follows:

• Two-compartment: The two-compartment inverse problem involved estimating two parameters (associated with the cytoplasmic and nucleic compartments); thus, the parameter set $\mathit{\gamma }=\left\{a_C,a_N\right\}$. The parameters were estimated in a two-step process: (i) first, the cytoplasm indentation was performed to estimate $a_C$, then (ii) the nuclear indentation was performed to estimate $a_N$.

• Three-compartment: In the extended three-compartment model (including the nuclear membrane), we introduced a third synthetic force-indentation dataset generated from forward simulation with the fitted two-compartment model, as this new information is necessary to make the parameter estimation problem well-posed [28]. The location on the cell at which the synthetic force-indentation curve was generated is visualized in Figure S1. Thus, the approach in the three-compartment model involved estimating the parameter set $\mathit{\gamma }=\left\{a_C,a_N,a_M\right\}$. The parameters were estimated in a two-step process: (i) first, estimating the cytoplasm parameter $a_C$ using the experimental force-indentation data acquired at the cytoplasmic location, and (ii) second, simultaneously estimating the parameters for the nucleoplasm and nuclear membrane ($a_N$ and $a_M$) by fitting the model to both the experimental nucleus force-indentation data and the additional synthetic force-indentation curve generated using the fitted two-compartment model.

Parameter correlations

The determinability of the material parameters was evaluated using calculations of the correlation matrix, as described in our previous study [37]. The correlation coefficient $C{R}_{pq}$, defining the correlation between the parameters ${\gamma }_p$ and ${\gamma }_q$, was approximated by [48]:

$$C{R}_{pq}\approx \frac{{\left({\mathbf{H}}^{-1}\right)}_{pq}}{\sqrt{{\left({\mathbf{H}}^{-1}\right)}_{pp}}\sqrt{{\left({\mathbf{H}}^{-1}\right)}_{qq}}};\text{no sum on}p,q,$$ (9)

where $\mathbf{H}={\mathbf{J}}^T\mathbf{J}$ is an approximation of the Hessian, or the information matrix, and $\mathbf{J}$ is defined as the sensitivity matrix with the components:

$$J_{np}=\frac{\Delta f_n}{\Delta {\gamma }_p},$$ (10)

in which $\Delta f$ and $\Delta \gamma$ are the changes in maximum reaction force and constitutive parameter, respectively, calculated at the optimal parameter values ${\mathit{\gamma }}^{*}$.

Analysis of well-posedness of the inverse problem

Parameter estimation from experimental data is an inverse problem warranting an analysis of whether it is well-posed. A problem is considered well-posed if a solution exists, is unique, and is stable with respect to small changes in the input data [49]. An ill-posed problem, in contrast, would suffer from non-uniqueness, where different parameter combinations could produce statistically indistinguishable simulated results, and from instability, where a small change in parameters could lead to drastic changes in the simulated results.

To analyze the well-posedness of our parameter estimation, we examined the Hessian of the cost function at the optimal solution (H). A positive-definite Hessian (all positive eigenvalues) indicates a distinct local minimum and a locally unique solution [49]. Conversely, a singular or near-singular Hessian (one or more zero or near-zero eigenvalues) is definitive proof of an ill-posed problem, as it indicates the existence of a set of non-unique solutions. Furthermore, the degree of well-posedness between different fits can be compared using the condition number of the Hessian:

$$K=\frac{{\lambda }_{\mathit{\text{max}}}}{{\lambda }_{\mathit{\text{min}}}},$$ (11)

where ${\lambda }_{\mathit{\text{max}}}$ and ${\lambda }_{\mathit{\text{min}}}$ are the largest to smallest eigenvalues of the Hessian, respectively. A low condition number signifies a robust, “round” minimum, indicating a stable fit. Conversely, a high condition number indicates an ill-conditioned problem that may be highly sensitive to noise in the experimental data.

2.3. Optimal design for elasticity estimation

In the context of AFM tests, there are infinite combinations of potential probing sites. Thus, obtaining experimental data from an optimal set of probing locations is critical to achieve minimal covariance among the unknown constitutive parameters. For practical purposes, we chose a discrete set of locations to evaluate the determinability of the constitutive parameters. Various groups of forward simulations were conducted, each with a distinct set of probing locations, and the optimal group was determined using the following method.

Optimal design criteria

The optimal set of probing locations was determined by minimizing the confidence region of the parameter set $\mathit{\gamma }$, effectively minimizing the covariance of the unknown parameters [37]. This process was accomplished using the D-optimality criterion, defined as:

$${\Delta }_D=\mathit{\text{det}}\left(\mathbf{H}\right).$$ (12)

The confidence region was minimized by maximizing the D-optimality criterion of the information matrix $\mathbf{H}$.

Two-compartment cell model

In the two-compartment model, three simulation groups were conducted. Each one consisted of two probing sites: one above the nucleus and one at a chosen distance $d_1$ from the nucleus site on the cytoplasm (Fig. 4). The D-optimality criterion for each of the three sets was determined.

Fig. 4.

(A) Two-compartment model: schematic indicating the probing locations evaluated in the optimal design approach for the two-compartment cell model. The nuclear probe is applied above the center of the nucleus, while the cytoplasmic probe is applied above the cytoplasm at a point at a distance $d_1$ from the nuclear probe. (B) Three-compartment model: schematic indicating the four parameters defining the locations at which the two AFM probes will be applied above the cytoplasm. The axis $x_1$ is defined as a line passing through the nucleus’ center and the furthest point along the border of the cell cytoplasm. The nuclear probe is applied above the center of the nucleus. The location of each of the cytoplasmic forces $F_n$ is defined by a distance $d_n$ from the center of the nucleus and an angle $\theta$ measured from the axis $x_1$.

Three-compartment cell model

In the three-compartment model, 10 groups of forward simulations were performed. Each consisted of three probing sites: one above the nucleus and two on the cytoplasm away from the nucleus. The probing sites on the cytoplasm were chosen as follows. First, one cytoplasmic location was chosen at a distance $d_1$ from the nuclear probing location. The line connecting the nuclear and first cytoplasmic locations was then defined as a reference axis for the second cytoplasmic location. The second cytoplasmic probing location was defined as a point at a distance $d_2$ from the nuclear location and an angle $\theta$ relative to the reference axis (Fig. 4). These parameters defining the sites for each of the ten experiments were chosen via Latin hypercube sampling (LHS).

Evaluation of the optimal parameter estimation

We then tested the performance of optimal and sub-optimal constitutive parameters (estimated at optimal and sub-optimal probing locations as identified by the D-optimality analysis) to assess the importance of utilizing an optimal set of probing locations. A critical test of the overall parameter estimation procedure was to assess how well the estimated parameters predict behavior at a location in the cell different from where the initial fitting occurred. After comparing the D-optimality criterion values for the two-compartment model, we generated force-indentation data from the cell model at the identified optimal and sub-optimal probing locations. We then repeated the inverse problem using both sets of probing locations, each resulting in a set of “optimal” and “sub-optimal” constitutive parameters. Finally, we used both optimal and sub-optimal parameters to simulate the force-indentation relation at different points on the cell surface. We evaluated the predictive performance of each parameter set by calculating the percent error between the ground-truth and predicted maximum reaction force at each probing location.

3. Results

We begin by presenting the results of the two-compartment inverse problem used to estimate the material properties of the HUVEC cytoplasm and nuclear compartment. We then present the results of the optimal design and the evaluation of the predictive capabilities of “optimal” vs. “sub-optimal” parameters. Next, we show the results of the inverse problem applied to estimate the properties of the nuclear membrane in the three-compartment model. The optimal design approach to the three-compartment model identified three optimal probing locations. Finally, we present the results of a parametric study investigating the effect of nuclear membrane thickness on AFM measurements. The analysis of the two-compartment model was conducted under purely hyperelastic conditions. The viscoelastic behavior was investigated in the three-compartment cell model.

3.1. Two-compartment cell model

Experimental data at nucleus and cytoplasm

The AFM experimental force–displacement response at both sites above the nucleus and far from the nucleus (above the cytoplasm alone) exhibited an exponential-type response, consistent with previously published studies of cellular mechanical behavior [23,50,51]. The apparent stiffness above the cytoplasm appeared to be greater than that above the nucleus (Fig. 5A).

Fig. 5.

Overview of the parameter estimation of the cytoplasmic and nucleic compartments. (A) The inverse modeling approach closely replicated the median force-indentation response in the two-compartment model. (B) The inverse problems conducted with the two-compartment cell model estimated two parameters for the cytoplasm $\left(a_C\right)$ and the nucleus $\left(a_N\right)$. Numerical values corresponding to each set are available in the Supplementary Materials (Table S2). Visualization of the strain response from the application of the probe to the (C) cytoplasm and the (D) nuclear compartment.

Cytoplasmic and nuclear compartment parameter estimation and well-posedness

The inverse parameter estimation for the two-compartment model (nucleus and cytoplasm) resulted in a set of $a$ parameters (Table S2) that provided a high-fidelity match between the model’s simulated force–displacement curves and the experimental data (Fig. 5A). We assessed the well-posedness of these inverse problems by evaluating the Hessian of the cost function at the solution. The inverse problem for the cytoplasmic parameter $a_C$ yielded a condition number of $K=1.25$, and the problem for the nucleus parameter $a_N$ yielded a value of $K=15.7$ (Table S3). This confirms that both parameters are estimated using a well-conditioned problem, providing a stable solution. Visualization of the strain distributions at the cytoplasmic and nucleus probing locations revealed local non-zero strains and displacements near the probing locations (Figs. 5C–D, Fig. S2).

Optimal design and evaluating predictive capability

We used the D-optimality criterion to compare the three sets of probing locations for the two-compartment model (Fig. 4A), with the results presented in Table S4. This analysis used the constitutive parameters estimated from the inverse problem (Table S2). Maximizing the D-optimality criterion indicates optimal parameter determinability and minimal parameter covariance. The numerical values clearly identified the optimal choice for robust parameter estimation: the combination of one probing site at the nucleus and the cytoplasmic probing location furthest from the nucleus (Fig. S3).

Next, we used the information gained from the D-optimality analysis to evaluate the predictive capability of two sets of estimated parameters: one set estimated with data from the optimal probing location (Table S4, Exp 1) and one set from data at the sub-optimal location (Table S4, Exp 2). We first generated synthetic force-indentation data at each probing location by running a forward simulation with constitutive parameters estimated via the two-compartment inverse problem (Table 1i). Then, two inverse problems were conducted using an initial guess for the constitutive parameters (Table 1ii): one at the sub-optimal and one at the optimal probing locations. The inverse problems thus generated a set of “optimal” parameters (Table 1iii) and a set of “sub-optimal” parameters (Table 1iv). Lastly, we gathered force-indentation data from forward simulations using both the optimal and sub-optimal parameter sets at matched alternate locations on the cell cytoplasm to evaluate the predictive capability of each parameter set. Inverse parameter estimation at the optimal probing locations identified parameters closer to the assigned set than those estimated at the sub-optimal locations. The use of optimal parameters resulted in a better prediction of the ground-truth force-indentation curve (Fig. 6A). This test also indicated that the simulated maximum force measurement using sub-optimal parameters was, on average, significantly lower than the ground-truth values, whereas there was no difference in maximum force between the ground-truth and optimal-parameter simulations. (Fig. 6B) (see Table 2).

Table 1.

Parameters estimated via an inverse problem in the two-compartment cell model at the optimal and sub-optimal probing locations identified via the D-optimality criterion (Fig. S3, Table S4). Parameters estimated at the optimal probing locations were found to be closer to the ground truth Assigned set than those estimated at the sub-optimal locations.

Parameter set	Cytoplasm ${\mathit{a}}_{\mathit{C}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$	Nucleus ${\mathit{a}}_{\mathit{N}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$
(i) Assigned set	6.87	3.66
(ii) Initial guess	5.0	5.0
(iii) Estimated at optimal locations	7.09	3.71
(iv) Estimated at sub-optimal locations	3.71	4.66

Fig. 6.

The robustness of constitutive parameters estimated using optimal and sub-optimal experimental design was evaluated by testing the predictive capability of those parameter sets at various locations on the cell cytoplasm; probing locations are shown in the inlayed image of the cell model. (A) A representative example of the force-indentation relationship at one of the probing locations shows that using optimal parameters more closely replicates the ‘ground-truth’ behavior. The representative force-indentation curve was acquired at the probing location identified with an asterisk. (B) The optimal parameters resulted in significantly less error between the ‘ground truth’ and predicted maximum force.

Table 2.

The chosen sets of probing locations (Fig. 4) considered in the optimality analysis with the purely elastic three-compartment cell model. AFM probing sites were selected using Latin hypercube sampling. The locations tested in Experiment 1 (green) and Experiment 2 (red) resulted in the greatest and lowest parameter optimality, respectively. Distance $d_i$ and angle $\theta$ are provided in units of μm and °, respectively.

	Force 1 ${\mathit{d}}_1\left(\mathit{\mu }\mathbf{m}\right)$	Force 2 ${\mathit{d}}_2\left(\mathit{\mu }\mathbf{m}\right)$	$\mathit{\theta }\left(°\right)$	${𝚫}_{\mathit{D}}\left({10}^6\right)$
Exp 1	12.7	15.3	27.8	543.6
Exp 2	11.7	15.3	112.4	9.53 × 10−22
Exp 3	11.7	19.1	52.6	0.795
Exp 4	9.33	9.44	19.8	2.07 × 10−2
Exp 5	21.1	4.64	52.5	0.242
Exp 6	6.21	2.62	9.4	10.6
Exp 7	10.3	6.60	67.5	8.34 × 10−2
Exp 8	3.03	22.4	20.5	8.47
Exp 9	20.3	3.74	64.4	6.97 × 10−2
Exp 10	3.89	11.0	88.4	0.22

3.2. Three-compartment cell model

The same inverse problem approach used for the two-compartment model was applied to an extended three-compartment cell model that included the nuclear membrane. The model’s hyperelastic parameters were estimated via an inverse problem, followed by an analysis of how viscoelastic behavior impacted the optimal design approach.

Nuclear membrane parameter estimation and well-posedness

First, the cytoplasm material parameter $a_C$ was estimated from the experimental force-indentation data at the cytoplasmic location. Next, we attempted to simultaneously estimate the nucleus $\left(a_N\right)$ and nuclear membrane $\left(a_M\right)$ parameters from using both the experimental nuclear force-indentation curve and a synthetic nuclear force-indentation curve generated from forward simulations of the two-compartment model (Fig. S1). The resulting material parameters are presented in Table S5, and a Hessian analysis confirmed that this inverse approach was formally well-posed, as the Hessian was positive-definite with only positive eigenvalues (Table S3). Correlation analysis revealed weak inter-material correlations (i.e., $a_C$ and $a_N;a_C$ and $a_M;a_M$ and $a_N$; Table S6). Importantly, the analysis showed that the membrane parameter is not highly correlated with neither the nuclear nor the cytoplasmic compartment parameter.

Identification and evaluation of optimal probing locations

The optimal parameter estimation for the three-compartment cell model was conducted using three probing sites, as three sets of experimental data are necessary for the well-posed estimation of three elastic moduli. This analysis was conducted under purely hyperelastic conditions; the impact of viscoelasticity was evaluated in a following section. The numerical comparison of different probing location sets in the three-compartment model (Fig. 4) is presented in Table 2. The D-optimality criterion clearly identified the force locations of Exp 1 as the optimal choice for robust parameter estimation. Conversely, Exp 2 yielded the lowest D-optimality value, representing the most sub-optimal design. The locations of the identified optimal and sub-optimal locations are portrayed in Fig. 7, and a visualization of the strain response when probing at the optimal (Exp 1) locations is provided in Figure S4.

Fig. 7.

Visualization of the (A) “optimal” and (B) “sub-optimal” sets of probing locations determined by the D-optimality criterion (Table 2).

We used the information gained from the D-optimality analysis to evaluate the predictive capability of two sets of estimated parameters: one set estimated with data from the optimal probing location (Table 2, Exp 1) and one set from data at the sub-optimal location (Table 2, Exp 2). Two inverse problems were conducted using an initial guess for the constitutive parameters (Table 3ii): one at the sub-optimal and one at the optimal probing locations. The inverse problems thus generated a set of “optimal” parameters (Table 3iii) and a set of “sub-optimal” parameters (Table 3iv). Inverse parameter estimation at the optimal probing locations identified parameters closer to the assigned set than those estimated at the sub-optimal locations.

Table 3.

Parameters estimated via an inverse problem in the three-compartment cell model at the optimal and sub-optimal probing locations identified via the D-optimality criterion (Fig. 7, Table 2). Parameters estimated at the optimal probing locations were found to be closer to the ground truth Assigned set than those estimated at the sub-optimal locations.

Parameter set	Cytoplasm ${\mathit{a}}_{\mathit{C}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$	Nucleus ${\mathit{a}}_{\mathit{N}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$	Membrane ${\mathit{a}}_{\mathit{M}}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$
(i) Assigned set	6.87	3.37	14.8
(ii) Initial guess	5.0	5.0	7.5
(iii) Estimated at optimal locations	7.07	3.59	15.4
(iv) Estimated at sub-optimal locations	7.08	3.78	12.6

Effect of nuclear membrane thickness and stiffness on force-indentation relationship

A parametric study of the effect of nuclear membrane thickness on the force-indentation relationship at the nuclear probing location indicated a linear relationship between membrane thickness and the maximum reaction force. A 4-fold increase in membrane thickness resulted in a ~4% increase in the maximum force (Fig. 8B). Similarly, an analysis of the effect of altered membrane stiffness on the force-indentation relationship at the nuclear site indicated a linear relationship between membrane stiffness and maximum reaction force. In this analysis, the ratio of the nuclear material constants $a_N/b_N$ was held constant. A 2-fold increase in membrane stiffness was found to result in less than a 1% increase in the maximum force (Fig. 8C), suggesting the force response is almost entirely dominated by the bulk properties of the nucleoplasm.

Fig. 8.

A parametric study was conducted to investigate the effect of nuclear membrane thickness and stiffness on maximum reaction force at the nuclear probing location measured via AFM. (A) Visualization of the stress distributions in a membrane of 200 nm thickness and $a_N$ of 14.8 kPa. The stress presented here is the Cauchy stress. The parametric study revealed a linear relationship between the maximum reaction force and both (B) the nuclear membrane thickness and (C) the membrane stiffness. In (B), maximum force was normalized to 21.03 nN; in (C) maximum force and membrane stiffness were normalized to 15.45 nN and 3.15 kPa, respectively.

Effects of viscoelasticity on optimal design

A parametric study of the effect of viscoelasticity on optimal design was performed using the three-compartment model. In these simulations, the cytoplasm and nucleoplasm were prescribed viscoelastic behavior while the nuclear membrane remained purely hyperelastic. Simulations were run using both high and low viscosity parameters defined by the relaxation constant $\eta$. For high viscoelastic behavior, ${\eta }_N=0.5$ and ${\eta }_C=0.0725$ for the nucleus and cytoplasm, respectively. For low viscoelastic behavior, ${\eta }_N=4.0$ and ${\eta }_C=0.2$ for the nucleus and cytoplasm, respectively.

Viscoelastic simulations confirmed that the choice of relaxation constant $\eta$ significantly impacts the time-dependant mechanical response. The results of the viscoelastic force-indentation simulations in the three-compartment model are illustrated in Figures S5A–B. These representative curves demonstrate the energy dissipation and hysteresis characteristic of the cell’s time-dependent mechanical response, and qualitatively match the hysteresis observed in the full loadingunloading experimental AFM curve for each cell compartment (Figs. S5C–D). Force-indentation curves for parameters chosen to represent low viscoelasticity show a predominantly hyperelastic response. The loading and unloading paths are nearly identical, resulting in a very small hysteresis loop, which indicates that stresses relax almost instantaneously (Fig. S5A). Contrastingly, the parameters chosen to represent high viscoelasticity produced a pronounced time-dependent effect. The corresponding force-indentation curve demonstrates a large hysteresis loop, representing significant energy dissipation and a much slower relaxation process (Fig. S5B). These two distinct cases establish clear “near-elastic” and “highly-viscous” bounds to evaluate the robustness of the optimal design of experiments approach.

The inclusion of viscoelastic behavior significantly altered the conclusions of our optimal design analysis. For the three-compartment model, the D-optimality scores differed not only numerically (Tables S7, S8) but also resulted in a shift in the relative ranking of the probing locations compared to the purely hyperelastic case. This change highlights that the optimal experimental design is intrinsically coupled to the constitutive behavior. A design that maximizes determinability for static equilibrium does not necessarily do so for time-dependent parameters.

4. Discussion

4.1. Improving parameter identifiability in multi-compartment cell mechanics

Cellular mechanobiology is a crucial research area with significant implications for understanding disease progression and designing treatments. The physical environment in which cells exist plays a critical role in governing numerous cellular activities, including growth, movement, specialization, and genetic activity. Gaining insight into how changes in intracellular mechanical conditions affect cellular behavior offers unique perspectives into disease mechanisms. For instance, recent work has highlighted the significance of altered cell compartment elasticity in mechanotransduction in senescent cells [9]. Their findings suggest that the cytoplasmic-to-nuclear stiffness ratio is critical for transferring shear stress from the cell’s exterior to the nucleus, highlighting the interplay between mechanical cues and biological responses. Moreover, recent studies have suggested mechanotransduction is dynamic, and cells self-regulate their mechanics in response to different environments [52,53].

Characterizing the distinct mechanical properties of subcellular compartments remains a persistent challenge in cell mechanics. One methodology involves the segmented analysis of single force-indentation curves, in which the initial, shallower slope is attributed to the cytoplasm, and subsequent, steeper slopes are assigned to stiffer underlying structures, such as the nucleus [54–56]. This interpretation has been combined with in silico validation and used, for example, to characterize actin depolymerization in the cytoskeleton [55]. A second approach combines AFM with real-time microscopy (e.g., light-sheet or spinning-disk confocal microscopy) to monitor subcellular deformation under load [57,58]. By correlating force application with the geometric changes of intracellular structures, these correlative methods have revealed that nuclear morphology, including volume and curvature, plays a critical role in mechanotransduction [58]. An alternative to these single-point or correlative methods is full-field force mapping using hundreds or thousands of indentations to generate a high-resolution elasticity map of the entire cell surface [16–18]. The wealth of data provided by this approach has been used to estimate nucleolar properties [20] and, when combined with the segmented analysis approach, has generated high-resolution images of subcellular structures and organelles, such as the nucleus or stress fibers [19].

While these methods provide valuable insights, they often struggle to yield unique and robust material parameters [25,27]. AFM data is often coupled with FE modeling to extract further understanding from experimental force-indentation data and to make predictions by enabling the inverse estimation of specific material parameters for complex, multi-compartment constitutive models [9,20]. However, AFM-FE approaches depend heavily on experimental design. For example, arbitrary probe locations may fail to capture the necessary information to decouple component properties. Even full-field force mapping, while data-rich, is not inherently optimal; it can generate vast amounts of redundant data that fails to decouple compartment elasticities and is computationally prohibitive for inverse analysis.

Our framework is designed to overcome these limitations using two key components: (i) high-fidelity, cell-specific 3D reconstruction and (ii) a formal optimal design of experiments method. This approach directly addresses a core challenge: inverse problems in multi-compartment cell mechanics are often ill-posed or poorly conditioned. This ill-posedness typically stems from high parameter correlation, which can arise from two distinct sources: the constitutive model itself, which may have inherently codependent parameters [46,47], or complex cell geometry, which can introduce additional codependence by confounding the mechanical response at different locations.

Our framework is expressly designed to mitigate the parameter codependence caused by geometry. The optimal design of experiments method, guided by the D-optimality criterion, systematically finds the experimental design that minimizes the parameter confidence region. Results from our two-compartment model clearly demonstrate this principle: optimal parameters yield higher predictive accuracy than sub-optimal parameters. This shows that one experimental design can yield more accurate information about mechanical behavior under varying loading conditions than another. In practical terms, our framework identifies the most “informative” probing locations, minimizing geometry-induced parameter codependence to yield the most stable and reliable parameter set possible.

4.2. Image-based experimental–computational approach to standardize multi-compartment elasticity characterization

Integrating computational simulations with high-fidelity imaging and experimental techniques holds promise for standardizing AFM techniques and enhancing the reliability and reproducibility of parameter estimation. However, this integration does not automatically solve the fundamental challenge of parameter identifiability. As our analyses have demonstrated, the estimated parameters range from highly robust to severely indeterminate. The “quality” of a parameter set can be quantified, for example, by the D-optimality criterion. Our analysis, which showed a dramatic improvement in parameter determinability between optimal and suboptimal sites, clearly demonstrates this.

The presented approach for image-based optimal mechanical characterization makes use of a generalizable method to systematically maximize the mathematical conditioning of the problem and deconvolute spatial geometric contributions from mechanics, thus minimizing variability and correlation between material parameters. Our findings further indicate that this optimization process is necessary even when considering the choice of constitutive model. Specifically, the optimal probing locations determined for the hyperelastic model proved suboptimal when viscoelastic behavior was incorporated, suggesting that the optimal experimental design is critically dependent on the time-dependent mechanical response of the sample. Therefore, for viscoelastic materials, the optimality assessment must rigorously account for the time-dependent constitutive behavior. This would necessitate the simultaneous estimation of key viscoelastic parameters (such as relaxation times or moduli ratios), which could be integrated as variables within the broader optimal design framework to fully characterize the energy dissipation behavior of the system [59].

The optimal design of experiments approach applied to parameter estimation via inverse modeling involves three main steps: i) an in silico inverse problem with AFM data at several sites on the image-based cell model, ii) assessment of D-optimality to identify the optimal number and location of probing sites and estimation of the parameters using an inverse problem approach, and (iii) validating the superior predictive capability of the optimal set over a non-optimal set of parameters. While this framework makes of use of super-resolution SIM for reconstruction of the cellular geometries, the framework is designed to be adaptable and can readily incorporate 3D cell models generated from any high-resolution imaging technique. This approach is grounded in maximizing the robust solution of the inverse problem as the D-optimality criterion is, in fact, mathematically linked to the Hessian of the cost function. It operates on a Hessian-like operator (the information matrix, derived from the sensitivity matrix) to find the experimental design that minimizes the volume of the parameter confidence region. In practical terms, this identifies the probing locations that are maximally “informative”, reduce parameter codependence, and yield the most robust, stable, and unique parameter set possible, even when operating within an inherently difficult or ill-posed problem.

While we have shown here that this rigorous optimal design framework, that leverages super-resolution microscopy and hyperelastic modeling can improve the robustness of parameter estimation, these advanced techniques require substantial instrumentation and computational resources, suggesting that for many routine applications, simpler analytical approaches may still offer a practical alternative. However, our results suggest that when the research question requires distinguishing between the mechanical properties of specific subcellular compartments (such as the nuclear membrane, nucleoplasm, and cytoplasm) and attaining a robust parameter set for predictive modeling, the enhanced accuracy achieved through our method is critical. The novelty of this work, therefore, lies not in the combination of imaging and FE modeling, but in the application of a formal optimal design methodology to transform ill-posed parameter estimation problems into more robust ones. In these cases, the potential uncertainty from using standard methods may be significant enough to obscure important biomechanical insights. Eventually, machine learning models trained on large datasets of biomechanical measurements and imaging data could further generalize and streamline this process, automating the identification of optimal probing sites and parameter estimation, while also making the methodology more accessible for a broader range of researchers and applications [60,61].

4.3. Improved predictive capability of multi-compartment cell models

Mechanical characterization of cells has proven invaluable for understanding how cellular structures adapt, and mechanotransduction is modulated in disease [9,62,63]. The mechanical behavior of cells is investigated using various experimental techniques [13]. Cantilever manipulation methods, such as AFM, are widely employed for their simplicity and accuracy. AFM excels in measuring forces at the piconewton scale on entire cells [25], compartments of the cell [21,23,64], and nanoscale cytoskeletal structures [25]. Despite its utility, studies indicate significant variability in AFM measurements based on probe shape and measurement location [25,26,65]. Additionally, multi-compartment biomechanical characterization within the same cell introduces increased parameter correlation and uncertainty. Indeed, our results showed an unsatisfactory correlation between parameters when using two AFM measurements to estimate the elasticity of three compartments.

Integrated experimental–computational approaches in cellular mechanobiology hold promise to improve the reliability of estimated parameters and the predictive capability of detailed cell models. However, attempts to include additional cell compartments in the elasticity characterization introduce further parameter uncertainty and covariance. This underscores the need for systematic characterization to ensure that the chosen parameter sets adequately reflect the underlying biological processes and that estimated parameters can competently predict cellular behavior across various physiological and pathological conditions.

The presented approach for optimal parameter estimation addresses these challenges by leveraging in-silico modeling to strategically identify experimental data that leads to reliable and reproducible parameters. Indeed, our results indicate that such an approach can improve predictive capability, allowing for more accurate simulations of cellular behavior under a broad range of loading conditions. Simulations in both the two-compartment and three-compartment model have shown that an optimal design of experiments approach to AFM can identify parameters that can more accurately predict mechanical behavior at locations different from those at which the original measurements were taken, thus improving the predictive capability of the model. By carefully selecting data points that capture key mechanical responses, our method enables the identification of parameter sets that can lead to consistent predictive power of cellular behavior across different cell types, experimental setups, and physiological states. Ultimately, this integrated experimental-computational approach enhances the ability to predict how cells will respond to various mechanical stimuli in vivo, potentially leading to the identification of novel biomechanical therapeutic targets.

4.4. Future directions and limitations

We introduced a robust approach for identifying AFM probing locations that produce a reliable set of material parameters. However, our method is subject to certain limitations. One limitation of the current study is the reliance on only two experimental measurements to characterize the mechanical properties of three cellular compartments. This limitation may have introduced additional uncertainty in the accuracy and precision of the parameters estimated for the nucleoplasm and membrane. Furthermore, the optimal probing locations identified by our computational approach have not yet been experimentally validated in real cells, which remains a key step toward ensuring the accuracy of our findings. Finally, the current model employs a discrete compartmental approach, neglecting the substantial heterogeneity within those compartments. Future iterations of the proposed optimal design approach could potentially resolve finer spatial variations in mechanical properties by increasing the density of the modeled sub-compartments.

In future work, we aim to experimentally validate the proposed optimal probing design to confirm its predictive capability on live cells. Additionally, leveraging this experimental–computational approach could enhance our understanding of mechanosensitive diseases, such as progeria and cancer, where cellular mechanical properties play a crucial role in disease progression. One avenue for translating this finding into direct experimental application is to train a machine learning (ML) model using both experimental and simulated data to identify optimal probing locations from imaging data, thereby eliminating the need for time-consuming and costly inverse problems. Such an approach would take into account the essential characteristics of the cell, including compartment-specific 3D geometry, to make a predictions. An ML process could make the presented approach more accessible and efficient for future studies on cellular biomechanics in health and disease.

This study introduced a novel experimental–computational framework to facilitate the accurate and reproducible characterization of previously understudied cellular compartments, such as the nuclear membrane. Studies have indicated that altered mechanotransduction is a crucial regulator of disease progression; thus, the presented method holds promise for more effective investigation of how the mechanical properties of cells drive behaviors such as malignancy in cancer and deregulation in aging cells. This improved understanding of the mechanisms of disrupted mechanosensing has the potential to accelerate the discovery of new therapeutic targets, ultimately leading to better, more personalized treatments for age-related conditions and cancer.

Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.cmpb.2026.109269.

Data availability

The data that support the findings of this study are available from the corresponding author, R.A., upon request.