An integrative workflow to infer viral and immune kinetics within human airway epithelium

To infer the complex, intermingled dynamics of viral and innate immune processes within human airway epithelium from standard experimental measurements, we developed a workflow that is able to simultaneously evaluate time-resolved bulk measurements and image data (Fig 1). By connecting time courses of viral load, IFN concentrations and tissue pathology, e.g., measured by the transepithelial electrical resistance (TEER), with spatially-resolved and individual cell-based measurements on infection patterns, we aim at retrieving the relevant information of local and global dynamics in order to assess the infection dynamics on a tissue level.

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Fig 1. Sketch of the complex dynamics and the developed workflow to infer viral spread within HAE:

With the complex dynamics of respiratory viral spread within HAE being influenced by (1) viral-specific transmission modes and infection kinetics, (2) tissue heterogeneity and regeneration dynamics, and (3) innate immune responses, an integrative approach is developed to infer and disentangle these dynamics by simulation-based Bayesian inference allowing the combination of multimodal data. Thereby, interference relies on two interconnected neural networks, i.e., including a summary (sumNN) and conditional invertible (cINN) neural network that are trained using simulated data.

https://doi.org/10.1371/journal.pcbi.1014248.g001

The approach relies on the concept of simulation-based Bayesian inference [31]. Similar to analysing viral dynamics by mathematical models, it requires the assumption of a mechanistic framework that provides a mathematical description of the assumed underlying processes [12]. To this end, we extended and adapted our previous multi-cellular and multi-scale cellular Potts model of HAE [22] to simulate the spatio-temporal dynamics of viral and immune dynamics on a tissue-level. From the simulations we can generate synthetic analogues of experimental measurements, including time-resolved standard population-level metrics for infection spread and tissue integrity (e.g., viral load, TEER), as well as image-based information that provide spatially-resolved measurements of infection patterns (e.g., abundance and distribution of clusters of infected cells) corresponding to histological images (S1 Text).

In contrast to likelihood-based (e.g., Maximum-Likelihood-Estimation) and traditional likelihood-free methods (e.g., Approximate Bayesian Computation [27]), parameter inference is based on the Bayesian neural parameter estimation framework BayesFlow [30]. The workflow consists of two interconnected neural networks, i.e., a summary and a conditional invertible neural network (cINN), that are jointly trained using simulated data [30,32]. By simulating the model M for various parameter combinations to generate synthetic data, a summary network is trained to learn characteristic summary features of these observed data, followed by the cINN to learn a bijective mapping between the parameter combinations and a latent distribution conditioned on the simulated observations. After the simulation-based training phase, the (inverted) cINN is then applied to the actual data, providing estimates of the Bayesian posterior distribution of given the observed data (Fig 1). The approach allows for multimodal data integration without the necessity to define or weight specific summary statistics to balance the consideration of local (spatial) and global (bulk) information (see also S1 Text, as well as [30] for further details on the method).

In the following, we validated the general applicability of our approach using models of increasing complexity to determine the ability of inferring kinetic parameters that describe infection spread by accounting for (1) viral-specific transmission modes and infection kinetics, (2) tissue heterogeneity and regeneration dynamics, and (3) innate immune responses (Fig 1). Thereby, we also examined how effectively kinetic parameters can be determined under different observational constraints across a broad range of infection scenarios.

Integrating spatial information is essential to infer viral transmission kinetics within tissues

In a first step, we evaluated the effectiveness of the simulation-based amortised Bayesian inference approach in determining viral infection kinetics within homogeneous tissues. To this end, we developed a spatially-explicit CPM implementation of a target-cell limited model, M_HOM, which represents a simplified version of the previously developed model [22], and only distinguishes between susceptible (S), infected (I) and infectious cells (J) (Fig 2A). Susceptible cells can become infected by either (i) cell-free infection dependent on the extracellular viral load (V) and a transmission parameter , or (ii) the direct contact to infectious neighbouring cells regulated by a transmission parameter , with the relative contribution of these two modes further controlled by f_cc. Upon infection, infected cells experience a viral eclipse phase determined by the transition parameter before they become infectious and start producing viral particles (). In this scenario, tissue pathology is mediated by the death of susceptible and infectious cells, without considering tissue regeneration (see also Materials and Methods and [22] for further details).

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Fig 2. Inferring viral kinetics in homogeneous tissues:

(A) Sketch of the processes of viral infection as considered in the M_HOM-model distinguishing between susceptible, S, infected, I, and infectious cells, J, and assuming homogeneous tissue conditions without tissue regeneration. (B) Progression of the infection and snapshots of the spatial distribution of infected cells for one particular parameter combination across n = 10 simulations. Measurements are obtained at 18, 42, 64, 90 and 114 h post infection. (C) Prior (grey bars) and posterior distributions obtained by the developed framework indicating inference of ground-truth parameters (dashed line) that control viral production, , transmission, , or transmission modes, f_cc, with (red bars) and without (blue) considering spatial image information in the inference procedure. The inference for one particular parameter combination is shown. (D) Simulated dynamics showing mean (solid line) and min-max prediction band across 100 simulations with the best parameter combination given the respective inference conditions. The synthetic data used for model analysis are shown in black (mean (dot) and standard deviation (whiskers)). (E) Global accuracy of parameter inference across 500 simulated ground-truth parameter sets for each of the three different parameters considered based on the Pearson correlation coefficient, R².

https://doi.org/10.1371/journal.pcbi.1014248.g002

The model M_HOM was then used to generate synthetic data that mimick experimental measurements, i.e., simulating the infection dynamics and obtaining bulk and image-based information, such as viral load, TEER and the distribution of infected cells at particular time points (Fig 2B). As we are particularly interested in the ability of inferring parameter describing viral kinetics, M_HOM was simulated given different combinations for the parameters governing viral production, infectivity and transmission, i.e., , while all other parameters were kept fixed (see S1 and S2 Tables). Hereby, and define scaling factors for the viral transmission and production rates, respectively, in comparison to the standard parameterisations adopted from [22]. We generated 10⁴ training samples with each relying on a unique parameter combination , which was simulated several times to account for disrupted sampling (S1 Text). Using simulation-informed prior distributions for the individual parameters assists the choice of the sample size to balance computational costs and simulated cell numbers (S1 Text).

With variations in the relative contribution of viral transmission modes, i.e., f_cc, showing strong effects on the size, abundance and spatial distribution of clusters of infected cells only visible within image data (S1 Fig), we specifically examined how the integration of spatially-resolved image information influences the ability of inferring viral infection kinetics. To this end, our Bayesian Inference approach was trained either with or without the spatial information on the distribution of infected cells provided by image-based analyses on the synthetic data. This included the average cluster size and abundance, the distance between individual clusters, as well as the average number of infected neighbours per cell, with details provided in S1 Text.

Training both networks on bulk and spatial information showed a remarkable ability in inferring the parameters describing viral spread within the tissue, given by a notable contraction in the posterior estimates compared to the prior distribution for each individual parameter (Fig 2C), which also includes parameters that were considered less frequent in the prior distributions used for training, i.e., considering rare infection patterns (S2 Fig). This could also be seen in the model capacity of re-simulating the ground-truth infection dynamics, with predictions being able to recover similar uncertainty as observed in the synthetic ground-truth data (Fig 2D). Global accuracy for all parameters given by the correlation between the ground-truth and inferred parameters across a validation set of 500 different parameter combinations not considered within the training phase reached R² > 0.9 (Fig 2E, — R² = 0.995; — R² = 0.925; f_cc — R² = 0.977). In contrast, neglecting spatial information leads to larger uncertainty, especially for the parameter f_cc describing local infection dynamics (Fig 2C-2E), with a decrease in the global precision to almost half compared to a situation including spatial information (). Thus, the presented approach is able to reliably infer viral infection kinetics within homogeneous tissues from experimental based measurements, with the integration of image-based information being essential to reliably infer local transmission kinetics.

Tissue heterogeneity and regeneration dynamics do not impair inference of viral kinetics

To further assess how cell heterogeneity and tissue regeneration influence the inference of viral infection kinetics, we repeated our analyses by using an extended model that accounts for the heterogeneity and structure of human airway epithelium, M_HAE. The model represents our previous CPM of HAE [22] and distinguishes between basal (B), secretory (S), and ciliated (C) cells that are organised within a pseudo-stratified epithelium. Cells are characterised by individual cell type-specific infection and viral production kinetics, with the model additionally accounting for tissue regeneration and cell differentiation pathways that follow proliferating basal to secretory and subsequently ciliated cells (Fig 3A, see also Materials and Methods, and [22] for a full description of the model and its implementation). In contrast to the original version [22], the impact of innate immunity is neglected at this point.

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Fig 3. Inferring viral kinetics in heterogeneous tissues.

(A) Processes and their characteristic parameters considered within the M_HAE-model resembling HAE by distinguishing between basal, B, secretory, S, and ciliated cells, C. Tissue regeneration and cell-type specific infection kinetics are additionally included. (B) Simulated viral spread in HAE with snapshots obtained at different time points. (C,D) Prediction accuracy of inferred parameter estimates describing viral kinetics, , using the BayesFlow-framework and including spatial measurements. Predictive checks for the cell and viral load dynamics for one particular parameter combination (C), and global accuracy of parameter inference across 500 synthetic ground-truth values (D) are shown. (E) Distribution of inference error for the M_HOM and M_HAE-model with and w./o. spatial information calculated by determining the L2-norm between inferred and synthetic ground-truth parameters across the 500 validation samples per condition. (F) Comparison of the generative performance of neural posterior estimators for M_HOM and M_HAE in resembling viral load and cell dynamics determined by the root mean squared difference across the simulated time series from 10 parameter combinations sampled from the posterior.

https://doi.org/10.1371/journal.pcbi.1014248.g003

Consistent with our previous analysis, M_HOM, we were interested in the ability to infer viral infection kinetics in these more complex conditions. In this case, the unknown infection kinetics were defined by , with and describing weighting factors that scale the cell type-specific viral transmission, , and production parameters, , , by maintaining their relative proportion. The individual processes of tissue regeneration and cell type-specific infection were parameterised using previous estimates obtained from experimental data of HAE cultures [22,33] (S2 Table). As before, the model system was used to generate synthetic data resembling experimental observations, with tissue heterogeneity leading to more variable spatial patterns of infection compared to M_HOM due to the cell type-specific infection dynamics (Fig 3B).

Applying our Bayesian inference approach trained on a large set of synthetic data to various test cases indicates a similar robustness in capturing the underlying dynamics and variation (Fig 3C), and global precision in parameter inference (Fig 3D) as observed for M_HOM across a large range of infection scenarios. Including spatial information in the analytical procedure was thereby essential to infer global and local infection dynamics, substantially decreasing the inference error (Fig 3E), which is determined by the L2-norm between inferred and given ground-truth parameters across the 500 synthetic validation samples, and the prediction error of inferred parameter combinations (Fig 3F) (S1 Text). Thereby, even infrequent and limited spatial information of just two time points is sufficient to allow a robust assessment of tissue kinetics, with intermediate and later time points seeming to be more informative when robust patterns have already emerged (S3 Fig). Thus, the efficacy of the BayesFlow inference approach in inferring the underlying viral transmission kinetics remains unaffected by the added complexity due to tissue heterogeneity and regeneration dynamics if these latter processes can be pre-determined.

Simultaneous inference of viral and innate immune kinetics requires additional prior knowledge

With the previous approaches neglecting innate immunity, we further assessed the ability to infer viral and innate immune kinetics from standard experimental measurements given the intermingled dynamics of these processes within tissues. To this end, the M_HAE-model was extended to account for the local auto- and paracrine effective dynamics of type-1 IFN signaling. IFN is assumed to be actively produced by infected and infectious cells with a cell type-specific parameter , with the local IFN concentration, , constantly degraded by . IFN acts by (i) inhibiting infected cells from becoming infectious and (ii) rendering susceptible cells refractory to infection, with both processes described by the effectiveness parameter . The transient protective state has an average duration of (see Fig 4A and Materials and Methods). This extended model accounts for antagonistic interactions of viral and immune dynamics within HAE culture systems that affect infection patterns, and also provides bulk measurements of the IFN-concentration (Fig 4B).

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Fig 4. Simultaneous inference of viral and innate immune kinetics:

(A) Sketch of the additional processes describing innate immune dynamics by IFN-expression within the -model accounting for autocrine and paracrine effective modes. (B) IFN load and fraction of cells refractory to infection for one simulated parameter combination . (C) Global accuracy of parameter inference based on the correlation of estimated and ground-truth parameters across 500 individual simulated parameter combinations using the Bayesian inference approach. Accuracy is shown for a scenario with an unknown (, left column) and known value (, right column) for the IFN–production rate, , with increased performance visible across all parameters in the latter scenario. (D) Combined and parameter-wise inference error showing the decreased uncertainty in parameter estimates for the innate immune kinetics in compared to .

https://doi.org/10.1371/journal.pcbi.1014248.g004

Evaluating the ability of the Bayesian inference approach to simultaneously infer viral and immune dynamics — described by the unknown parameters and f_CC for viral infection kinetics as before, and and for IFN production and effectiveness — we found that parameter inference is impaired even given spatial information on the infection pattern (Fig 4C). Based on the additional complexity and feedback dynamics, global accuracy of individual parameter inference decreased ranging from across all parameters. Especially the identification of parameters describing innate immune kinetics was impaired (Fig 4C). However, parameter inference for immune effectiveness could be restored if prior knowledge on IFN-production can be obtained. Pre-defining the value of , model leads to decreased uncertainty in parameter estimates (Fig 4D). Thus, additional bulk measurements of the total IFN concentration are not sufficient to simultaneously infer cell-based viral and immune dynamics without prior knowledge on specific aspects of the innate immune kinetics.

Analysis of SARS-CoV-2 infection dynamics within HAE indicates dominant cell-associated transmission modes

Having validated the general ability of the proposed approach in inferring viral infection kinetics on a tissue-level, we applied the method to experimental data on the infection dynamics of SARS-CoV-2 within human airway epithelium. Reconstituted epithelia of human primary airway epithelial cells obtained from one single donor were infected with SARS-CoV-2 and followed up to 72h post infection. At 0, 24, 48 and 72 hpi bulk and image measurements on viral load, tissue integrity and interferon concentration, as well as spatial patterns of viral infected cells were obtained (Fig 5A). Data indicated a continuous increase of viral load and the percentage of infected cells over the observed time period, reaching a considerable loss in tissue integrity at 72 hpi.

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Fig 5. Application to experimental data:

(A) Experimental protocol for analysing SARS-CoV-2 infection dynamics within HAE culture systems obtaining bulk and image measurements from HAE cultures of a single donor at 10, 24, 48 and 72 hpi, with 5 replicates per time point. (B) Experimental measurements (grey dots, mean±SD - black dots) and model predictions adapting model to the experimental data (red lines/ shaded area) for different characteristic quantities (see also S4C Fig) and Materials and Methods). (C) Snapshots of the infection patterns at indicated time points of the experimental data and representative model simulations. (D) Parameter estimates showing posterior contraction (red) from the prior (grey) for the four parameters estimated. (E) Reference curve relating f_cc to the actual fraction of infected cells by cell-to-cell transmission based on model predictions. Our estimates for f_cc for SARS-CoV-2 predict 84% [58%,100%] of the infections being due to local transmission dynamics.

https://doi.org/10.1371/journal.pcbi.1014248.g005

Adapting and re-training our model to the experimental protocol (see Materials and Methods, and S4A and S4B Fig), model predictions show good agreement with the observed dynamics for the various quantities considered, except for underestimating the infection load and tissue destruction at 72 hpi, which is characterized by a sharp increase between 48–72 hpi (Fig 5B). This also applied to the characteristics of the spatial infection patterns observed (Figs 5C and S4C). The analysis showed considerable parameter contraction for the estimated parameters affecting viral production, transmission and IFN efficacy (Fig 5D), with estimates and their 95% credibility intervals provided in S3 Table.

With our model able to retrieve the particular transmission mode by which a cell got infected, we determined a reference curve relating f_cc to the actual fraction of cells infected by cell-to-cell transmission. Our estimated parameter range for f_cc indicated that 84% [58%,100%] of infections were due to cell-to-cell transmission, pointing towards local transmission dynamics as the dominant mode of SARS-CoV-2 spread within heterogeneous human airway epithelium (Fig 5E).

Mathematical models

We used three different mathematical models of increasing complexity to determine and validate the ability of inferring viral and immune kinetics by our workflow. All models relied on the cellular Potts modeling (CPM) framework developed in [22] to simulate viral infection dynamics in HAE-culture systems using the Morpheus software [16]. In the following, the general processes and parameters considered within each of the different models are introduced, with the specific implementations of the stochastic dynamics within the CPM explained in detail below.

The target cell limited model for homogeneous tissue - M_HOM

As a first approach, we considered a simplified target cell limited model, M_HOM, to describe viral spread within a homogeneous tissue. The model distinguishes between susceptible, S, infected, I, and infectious cells, J. Susceptible cells can become infected either by diffusing viral particles dependent on the viral concentration, V, and a transmission parameter , or by cell-to-cell transmission via direct contact to infectious cells dependent on a transmission parameter , respectively. The probability to get infected by either of the two transmission modes is controlled by f_cc, which controls for the relative contribution of cell-free and cell-to-cell transmission to the infection dynamics (see below). After an eclipse phase with an average duration of , infected cells will become infectious and start producing viral particles by a viral production rate . The loss of susceptible and infectious cells is characterised by the parameters and , respectively, with the viral concentration V being reduced by the clearance parameter, c. For simplicity, proliferation of susceptible cells, as well as the loss of infected cells is neglected within this model. A sketch of the model is shown in Fig 1A. The following system of ordinary differential equations provides a mean-field approximation of the considered dynamics:

(1)

However, please note that the actual dynamics follows a spatially explicit and stochastic implementation as outlined below.

Modeling heterogeneous tissue dynamics - M_HAE

As an extension to the simplified M_HOM-model and to account for tissue heterogeneity, we used the human airway epithelium model, M_HAE, as it has been developed by [22]. For a detailed description of the model please see [22]. In brief, the model distinguishes between basal, B, secretory, S, and ciliated cells, C, considering their individual turnover and epithelial regeneration. Basal cells continuously differentiate into secretory cells which further differentiate into ciliated cells. Each cell type is characterised by cell type-specific infection dynamics, regulating viral transmission (), duration of the eclipse phase () and viral production () with the mean-field approximation for each cell type described analogous to Eqs. (1) (see also Fig 2A). In contrast to the model presented within [22], refractory cell types were not included within the M_HAE-model, as innate immune protection was neglected at this point.

Incorporating innate immune dynamics -

To fully mimick the situation of viral spread within HAE-culture systems, we extended the M_HAE-model to account for innate immune dynamics mediated by interferon (IFN). Upon infection, cells are assumed to start actively producing IFN after a lag-phase determined by an average duration with a cell type-specific parameter . The IFN concentration, , rendering susceptible cells to become refractory to infection, and inhibiting infected cells from becoming infectious. Dependent on the ratio between and , infected and infectious cells will be actively producing IFN. The protective cell state is assumed to be transient, with an average duration defined by , while the local IFN-concentration is degraded according to the parameter . A sketch showing the major processes of the IFN-dynamics considered within the model for one particular cell type is shown in Fig 3A, with the corresponding mean-field approximation described by extending Eqs. (1) to

(2)

For simplicity of notation, Eq. 2 considers , i.e., only infectious cells are producing IFN. This assumption is neglected in the stochastic, individual cell-based implementation below. Details with regard to the implementation of the different processes are explained below.

Cellular Potts model implementation in Morpheus

The different models were implemented within the cellular Potts modeling (CPM) framework Morpheus, a multi-scale and multi-cellular modeling environment allowing to simulate the spatio-temporal dynamics of individual cells [16]. The CPM is a lattice-based model with each cell comprising several grid sites that dynamically change their shape by aiming to reach and/or maintain a predefined target volume and surface area controlled by the parameters volume strength, V_S, and aspherity, . The basic model for human airway epithelium as presented by [22] (https://morpheus.gitlab.io/model/m6296/) was adapted to the different scenarios of the M_HOM, M_HAE and -model. In brief, the model considers a monolayer of cells (M_HOM) or pseudo-stratified epithelium (), with the latter achieved by allowing basal cells to be “overgrown” by secretory and ciliated cells due to a lower volume strength V_S for the former cell type.

To simulate the stochastic dynamics of the various processes within the different models we assumed a constant hazard model, with the specific parameters, as, e.g., for cell proliferation, differentiation and death, used to calculate the probability of an event within a time step of the simulation. For example, the probability of a cell to die within a discrete time step is calculated by . For cell death, cell differentiation and the duration of the viral eclipse phase, we additionally accounted for k = 4 intermediate compartments to simulate -distributed dwell- and lifetimes. Please refer to [22] for more details on the implementation. In the following, we will particularly explain the implementation of viral infection and immune dynamics, with the consideration of innate immunity deviating from the original implementation in [22].

Viral infection

Viral infection was implemented as previously described in [22] distinguishing between the two primary transmission mechanisms, i.e., cell-free and cell-to-cell infection. For cell-free infection, viral particles are produced at each grid site of infectious cells, J, at a constant rate . These particles can spread throughout the system, modelled by a diffusive field of extracellular viral particles, with the effective viral diffusion rate set to [22,24]. The probability of a cell i to get infected within a time period is then determined by , with V_i denoting the total concentration of viral particles across the cell body.

For cell-to-cell transmission, the probability of a cell to get infected, p_cc, is determined by the number of adjacent infectious cells, N_j, and calculated by . Hereby, the relative infectivity parameter controls the transmission efficiency of the infecting cell (i.e., comparable to the viral production), and the infectiveness of the receiving cell. The index x denotes the different cell types (i.e., basal, secretory, ciliated) when accounting for tissue heterogeneity.

In addition, a scaling factor f_cc is considered, which adjusts the relative contribution of each mode to viral spread. This leads to the overall transmission parameters and , with w_cf and w_cc denoting two weighting factors to adapt inferred bulk infection kinetics to the spatially-explicit modeling framework (see [22]). Please note that the values for , , and are considered cell type-specific and parameterised according to [22] for M_HAE and , while using the parameterisations for ciliated cells in M_HOM (see also S2 Table).

In order to test various viral kinetics, we additionally considered two scaling factors, and that scaled the cell type-specific rates and , respectively, while maintaining the relative ratios of cell type-specific viral production and transmission. Thus, the unknown parameters addressing viral kinetics in the different models are defined by .

Innate immunity

To consider innate immune response dynamics, the M_HAE-model was extended by considering the production and secretion of IFN by infected and infectious cells. Upon infection, infected cells experience a lag-phase and randomly progress towards an IFN-productive state with probability , considering k = 4 intermediate compartments that lead to -distributed lag-times. During this time, infected cells can also turn infectious, which in theory allows infected and infectious cells to be IFN-productive dependent on the ratio of and . Actively IFN-producing cells will produce IFN with a constant production rate , increasing the local extracellular IFN concentration, , at all grid sites covered by the cell. The IFN-concentration, , is modelled as an extracellular field of IFN particles which diffuse trough the simulated epithelium at a diffusion rate, , which was set to , based on previous analyses indicating a 10-times higher diffusion than for virions [37,53].

To account for the anti-viral and protective effects, i.e., autocrine and paracrine modes of IFN-signaling, the local IFN-concentration , which we use as a surrogate for IFN-signaling strength, influences (i) the viral production rate of infectious cells, (ii) the probability of infected cells to turn infectious, and (ii) the susceptibility of uninfected cells to infection. For each infectious cell J_i, the viral production rate is reduced dependent on the local IFN-concentration across the cell, , according to a Hill-function given by

(3)

Hereby, IC₅₀ denotes the IFN-concentration required for inhibiting half of the maximal viral production rate , and the Hill coefficient. To account for the paracrine mode of IFN-signaling, susceptible and infected, but not yet infectious cells can turn into a refractory state, R, which prevents them from getting infected or becoming infectious, respectively. The probability of turning refractory also depends on the local IFN-concentration, , and is calculated by with defining a transition parameter. The refractory state is assumed to be transient with average duration , and refractory cells become susceptible again with probability . Please note that for loosing the refractory status, we also use the concept of compartmentalisation, i.e., assuming k = 4 intermediate steps before making the transition to allow for -distributed transition times (see above). To account for the antagonistic effect of the viral load on the IFN-signaling capacity, the degradation rate of IFN is linearly dependent on the logarithm of the viral load at the corresponding grid sites with . In addition, the kinetics of interferon production or transition into refractory states is assumed to be not cell type-specific.

Default model parameterization

In contrast to the CPM modelling framework shown in [22], which mimicked a radial cell-culture system, each of the different models (M_HOM, M_HAE and ) was implemented in a rectangular grid with periodic boundary conditions. This was done for simplicity without loss of generality and transferability to actual experimental data. Each system considers a total number of 10⁴ cells in an area of ~1.163 mm² given an average cell density of cells, similar to the cell density used in [22]. In order to account for the changed structure of the model, the initial viral load, V₀, was adapted, starting each simulation with viral copies per . As before, simulations were performed with a time-step size of h (=15 min) and simulating the dynamics for 500 time steps, i.e., 125 hours in real time. Infection was initialised after a burn-in phase of 6h to let the tissue reach a steady-state. All parameters relating to cell type-specific cell sizes, i.e., volume strength and aspherity, cell turnover and differentiation, as well as viral infection kinetics were referred from [22], i.e., being largely SARS-CoV-2 specific (S1 and S2 Tables). For the model M_HOM, a monolayer of ciliated cells was assumed.

Inference of viral kinetics and innate immune dynamics focused on the parameters dependent on the particular model considered. The full list of parameters describing the general structure of the CPM and the model-specific parameters are given in the Supporting Information (S1 and S2 Tables, respectively). All simulations were performed within Morpheus version v2.3.3 with the source code of the model available at https://morpheus.gitlab.io/model/m6296/ and https://github.com/GrawLab/SARS-ALIculture.

Parameter inference by BayesFlow

For parameter inference of immune and infection kinetics combining time-resolved spatial and bulk measurements, we used the Bayesian neural parameter estimation framework BayesFlow [30]. The approach relies on Bayesian posterior estimation by a combination of different neural networks and uses model-based training to allow inference of the posterior distribution of an unknown set of parameters from a given data set. BayesFlow represents a computationally efficient method, as, e.g., in comparison to approximate Bayesian computation [27], and refrains from the necessity of defining appropriate summary statistics. Here, we briefly introduce the approach with detailed explanations of various steps given within the Supporting Information (S1 Text). For full details on the theoretical aspects and the implementation of BayesFlow, we refer the reader to the original publication [30].

The BayesFlow architecture consists of two modules comprising two jointly trained neural networks: (i) a sequence network module which combines a filtering network based on a 1D-convolutional layer and a long-short term memory (LSTM) recurrent network which provide a summary network that condenses high-dimensional observational data into a fixed-length summary vector, i.e., summarising characteristic spatial and bulk information dependent on the given kinetics; and (ii) an inference module which relies on a conditional invertible neural network (cINN) to learn a bijective mapping between parameter sets and a latent space conditioned on the observational data.

BayesFlow relies on simulation-based training. During the training phase, both networks are jointly trained by model simulations for a large variety of parameter combinations , creating similar output as expected from possible experimental measurements (see S1 Text). Thereby, the invertible neural network maps latent variables from a normal distribution to the parameters conditioned on the simulated observations X_i. This enables the network to learn the joint distribution . In the subsequent inference phase, the trained network is used to approximate the posterior distribution over parameters given real observational data x_obs, that is, . Owing to the amortized nature of BayesFlow, this inference step is fast and does not require re-training, provided that the data have the same structure as those used during training. Unlike traditional likelihood-free methods, such as approximate Bayesian computation (e.g., pyABC) [27], BayesFlow enables rapid inference after the initial training phase. This makes it especially well-suited for applications involving repeated inference tasks, such as estimating kinetic and immunological parameters from diverse datasets addressing viral infection dynamics.

The generation of training data, the specifications of the BayesFlow network architecture, as well as the methods for assessing and qualifying parameter inference as used within this study are outlined in detail in S1 Text. The prior distributions and calibration curves, as well as hyperparameters for the BayesFlow architectures for each of the different models considered are shown in S5-S8 Figs and S4 Table.

Experimental data

Parameter inference for experimental data

The parameter inference workflow was adapted to the experimental data, using the framework and prior information of the -model for generating training data. Snapshots of the simulation were captured at 10, 24, 48, and 72 hpi, using random crops representing approximately 2500 cells to mimic the number of cell nuclei observed on average in the microscopy data. To account for a possible selection bias during image acquisition that would favour selecting regions of infected cells, random crops with at least one infected cell cluster were considered in the training data. The derived spatial-level metrics were combined with measurements of viral load, IFN load, and TEER. For comparison with simulation data, the IFN load was converted using a factor of 2.751 × 10⁴ particles/pg mL⁻¹, assuming a 2.5% efficiency in antiviral induction. TEER values were normalized to the highest measurement.

The new inference networks were trained on n = 10⁴ synthetic datasets, followed by hyperparameter tuning with the architectures for both, the summary and cINN, adjusted to enhance the inference performance (see Supporting Information S1 Text). To assess the inference accuracy of these networks, simulated posterior retrodictive checks were performed in a two-step process. In a first step, 10 parameter combinations were sampled from the estimated posterior distribution to generate disrupted synthetic datasets. Model fits were evaluated by calculating the root mean squared distance (RMSD) between simulated and experimental data. The top 10 network architectures were selected and considered in a second retrodictive check that involved the sampling of 100 parameter combinations from each estimated posterior, with 20 simulations performed for each selected combination. Mimicking the experimental data, individual data sets were generated by randomly sampling from different parameter combinations and replica, resulting in five random replicates per time point and measurement. The process was repeated 100 times to generate 100 distinct datasets for each network architecture. The best fitting network architecture was selected by calculating the mean RMSD between simulations and experiments across the 100 disrupted datasets, and evaluating the network bias using the AUC-ECDF metric. To estimate cell-to-cell infection events during SARS-CoV2 infection in the experimental data, a calibration curve was generated to determine the relationship between the parameter f_CC and cell-to-cell infections. This involved sensitivity analysis of the f_cc parameter, keeping the other parameters constant across 100 simulations with 10 replicates each, followed by fitting a power law function. The estimated f_CC posterior distribution values were then used to calculate the fraction of cell-to-cell infection events given the calculated power law.