2.1 Ethics statement

The patient study was approved by the ethics committee of Tours University Hospital and was conducted in accordance with the Declaration of Helsinki. It was registered at ClinicalTrials.gov (NCT01895764).

Patients signed an informed consent to participate in the clinical study and agreed their data to be used for research purposes by authorized persons in accordance with the law. In the informed consent, the patients declared they acknowledge their data could be analysed by authorised persons by the promoter and by persons mandated by the Health authorities.

2.2 Study design

Patients fulfilling the Assessment of Spondyloarthritis International Society criteria were recruited between March 2013 and October 2014 within the HUGO network (Hôpitaux Universitaires du Grand Ouest-Western France University Hospitals) were enrolled in CoMARIS, a 26-week prospective, randomised, open-labelled, multicentre study [9]. All patients received adalimumab subcutaneously every other week, and half of them were randomly assigned to receive MTX subcutaneously 2 weeks before adalimumab (week ) as well as every week after that.

Data were collected from 110 patients, from whom measurements of adalimumab concentration and antibody titres were taken across five hospital visits (week 0, 4, 8, 12, and 26) [9]. Participants were drawn blood immediately prior to an adalimumab injection. However, 18 patients missed one or more visits, leaving a total of 91 patients with complete data used in the model calibration. Among these 91 patients, 49 received only adalimumab (whom we refer to as MTX-negative or MTX–), while 42 received adalimumab in combination with MTX (whom we refer to as MTX-positive or MTX+).

2.3 Biological analysis

ADA are detected with an antigen-binding test performed as described previously [7]. Antibody concentrations are compared with a standard serum containing ADA concentrations and expressed in arbitrary units per millilitre (AU/mL). A patient’s ADA-positive and ADA-negative status is classified respectively as titres above 12 and below 12 AU/mL at week 26. Taking into account all visits, ADA-low and ADA-high status is classified respectively as 13–100 and >100 AU/mL at any time as described previously [9].

Adalimumab concentration is measured by using a validated ELISA as indicated in [17]. Unfortunately, few TNF measurements are detected at baseline and due to the small sample size, they are not used in the model calibration (Fig D in S1 Appendix). Concentration measurements of MTX and of its metabolites (MTX-polyglutamate) are not provided.

In addition to the antibody and adalimumab measurements, lymphocyte analysis was performed for a subset of the 91 patients. Blood samples were collected across several visits (week -2, 4, 12, and 26) from 20 patients to analyze B and T lymphocyte counts. 11 of the remaining 19 samples are from MTX– patients (of whom 3 were classified as ADA-low (2 male, 1 female), and 2 as ADA-high (both female)). Finally, 8 of the 19 samples are from MTX+ patients (of whom 7 were classified as ADA-negative and 1 as ADA-high (female)).

Peripheral mononuclear blood cells (PBMCs) were isolated from blood collected in EDTA tubes using the Ficoll density gradient method (Ficoll-Paque PLUS, density 1.077g - GE Healthcare) and frozen until T cells and B cells phenotyping by flow cytometry. After defrost of cells, blocking was performed by adding 25 L of 1/20 human plasma to prevent non-specific antibody binding and then incubated for 20 minutes with fluorochrome conjugated antibodies as followed: for T phenotyping: anti-CD127- PerCpCy5.5, anti-CD25-PE and anti-CD4 FITC; for NK/T phenotyping: anti-CD3 APC, anti-CD4 FITC, anti-CD8 PerCpCY5.5, anti-CD16 BV510, anti-CD56 PE/Cy7; and for B phenotyping: anti-CD19-BV510, anti-CD24-PE, anti-CD27-PE/Cy7, anti-CD38 APC-H7, anti-IgD APC, anti-CD5-PERCPCY5.5, anti-IgM-FITC (all antibodies from BD Pharmingen). Dead cells were excluded by adding DAPI before acquisition on the FACSCanto II (Becton Dickinson).

2.4 Data description

In the group of 55 MTX– patients were 29 patients who developed ADA, three of whom had transient immunogenicity. Of these, 7 patients had high levels of antibodies in the final week 26. In contrast, only 14 patients in MTX+ cohort developed ADA, among whom 3 patients were classified as ADA-high.

From the patient visits data on several immune cell subsets, including naive T cells, identified as CD4 + CD25-, activated T cells (CD4 + CD25 + , however, we note that regulatory T cells also express the CD25 marker), NK cells (CD3-CD56+) as well as several subsets of B lymphocytes, including naive B cells (identified as CD19 + CD27-), memory B cells (CD19 + CD27+), and precursors to B regulatory (B10) cells (CD24^highCD27 + , CD24^highCD26^high) are available for calibration of the mathematical model.

There were no data available on memory CD4 + T cells or effector T cells, and monocytes, including macrophages, dendritic cells, and plasma cells were not measured. Importantly, the CoMARIS data set includes total counts of these subsets of immune cells, and no analysis of their ability to specifically recognise adalimumab has been performed. Therefore, the available data have been used as indicator of total cell count. All biological analyses were performed after study completion without the knowledge of clinical data or group of randomisation.

A comparison of the two treatment cohorts reveals some notable differences. The ADA titres are lower in the MTX+ cohort, while the adalimumab concentration was higher in this group, including among those who developed ADA [9].

2.5 Formulation of the mathematical model

We adapt a model for immunogenicity against therapeutic compounds [14] to the CoMARIS setting of anti-TNF- therapy based on adalimumab, whereby two patient cohorts are considered – with or without co-administration of MTX. Our model’s structure is simplified compared to that of the original model because the CoMARIS data set does not contain data on MHC-II alleles, B- and T-cell epitopes or type of ADA against adalimumab. Nevertheless, we could include additional types of immune cells available for the CoMARIS data set (NK and B10 cells) in the model. The basic scheme of the model is shown in Fig 1 and the variables are listed in Table 1. A system of ordinary differential equations describes the dynamics of TNF-, antigen-presenting cells (APC), CD4⁺ T lymphocytes, NK cells, B lymphocytes, adalimumab, MTX and antidrug antibody.

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Fig 1. Scheme of the model for the interactions between APC, T lymphocytes, B lymphocytes, adalimumab and MTX.

Dashed arrows denote activation, continuous arrows denote proliferation (if circling to the same compartment) or differentiation. Wavy arrows denote production of TNF-, or IL-10, up- and down-arrows following TNF-, MTX or IL-10 denote stimulation and suppression of the process.

https://doi.org/10.1371/journal.pcbi.1014740.g001

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Table 1. List of model variables.

https://doi.org/10.1371/journal.pcbi.1014740.t001

2.5.1 PK models (adalimumab, methotrexate, TNF-, ADA).

For the therapeutic monoclonal antibody adalimumab and for MTX we employ a two-compartment pharmacokinetic model (depot-central compartment). Adalimumab is administered in the depot (site of injection) as a pulse at a regular interval in equation (1) starting from week 0 (t = 0 days). Its uptake dose 40 mg corresponds to a concentration of mg/mL or 50000 g/mL. Adalimumab is absorbed into the blood at rate , and cleared from the blood at rate .

The volume ratio between the two compartments, denoted by , is estimated from the CoMARIS data, as shown in Table B in S1 Appendix. Adalimumab binds to its target, TNF- at rate , or to the antidrug antibody (ADA) at rate k₁₀₁ A U. However, as the binding to its target has a negligible effect on the adalimumab concentration, with [4] and TNF- barely measured at baseline in the CoMARIS data set, we neglect the term in the equation (2)

(1)(2)

Similarly, MTX (10 mg) is co-administered in the depot with an initial dose of mg/mL at days, followed by subsequent injections at an interval of days from day 0 as described [9]. MTX is absorbed into the blood at rate , and cleared from the blood at rate (equations (3)-(4)).

(3)(4)

Free antidrug antibody (ADA, equation (5)) is produced by both short-lived and long-lived plasma cells at a rate and undergoes clearance at rate .

(5)

Levels of ADA against adalimumab were reported also in other inflammatory diseases (for IBD the cut-off value for undetectable antidrug antibodies is 10 ng/mL [18]).

TNF- is a pro-inflammatory cytokine produced by various cell types: neutrophils, macrophages, dendritic cells, activated CD4 + T cells, NK cells [6,19,20], which modulates the inflammatory process and the immune response. Reported levels of TNF- in AxSpA are in the range pg/mL [21,22], but as already mentioned, sufficient data on TNF- is not available in CoMARIS (Fig D in S1 Appendix). In our model we assume a term for the background production of TNF, and specific production rates are given for TNF production by APCs, activated CD4 ⁺ T lymphocytes and mature CD3-CD56 + CD16 + NK cells respectively. TNF undergoes decay at rate , as well as inhibition by the binding to anti-TNF monoclonal antibody adalimumab, as reflected in the last term of equation (6).

(6)

TNF acts on different immune cell populations in a dose-dependent manner which we model as a Michaelis-Menten term with EC50 , and maximum rate depending on the context.

2.5.2 Antigen-presenting cells.

Antigen-presenting cells (APC) are crucial for the process of immunogenicity because they pick, process and display antigen peptides on their MHC-II complexes to activate the adaptive immunity. Unfortunately, APCs were not measured in CoMARIS since APCs localise in lymph nodes, tissues and sites of inflammation. As proxy for APC counts we use a combination of dendritic cells (DCs) and macrophages. DCs always express MHC-II, but unlike DCs, macrophages do not always express MHC-II. Only after IFN-induced activation these cells synthesise MHC-II.

At homeostasis for MTX-negative patients, resting APCs are produced at a rate equal to their removal rate and have a carrying capacity (7). Resting APCs uptake of antigen (adalimumab) at rate and become antigen-loaded APCs (8) before they enter the mature APC stage D₁. TNF is important for maturation and survival of DCs as evidenced by impaired functionality of DCs derived from rheumatoid arthritis patients on anti-TNF therapy [23] and by NF-B activation post TNF stimulation [24]. In (8) we model the TNF -mediated pro-maturation signal as an increase of the maturation rate by up to 50% for high levels of TNF. In (9) we model the TNF-induced survival of APCs as a reduction in the removal rate of mature APCs [25], so that at high levels of TNF the removal rate can be reduced by up to 50%.

Experimental data [10,26,27] reveal that MTX exerts an inhibitory effect on monocytes and macrophages. The suppressive action of MTX on APC activation is included in the equations for resting APCs in a dose-dependent manner at rate . Finally, the suppressive action of IL-10 on the maturation of APCs (for example, on dendritic cells [28,29]) is included indirectly using the relative change in B regulatory (B10) cell population, via a proxy term in (9), where . This is a reasonable assumption that does not require much additional complexity in the model because B10 lymphocytes secrete rapidly large amounts of IL-10, and are able to downregulate the immune response in inflammatory diseases solely via IL-10 [30]. Even though other immune cell populations such as monocytes or Th2 cells produce IL-10, this response may be context dependent and the lack of data on IL-10 or Th2 cells from CoMARIS makes the assessment of additional sources difficult to quantify.

(7)(8)(9)

2.5.3 CD4 + T lymphocytes.

The activation of naive and memory CD4⁺ T cells depends on the relative abundance of APCs available due to the limited number of T lymphocytes with which an APC can react at a given time. We use saturation terms of the form proposed by [14] to model this assumption: for the activation of naive CD4⁺ T cells in equation (11) and for the activation of memory CD4⁺ T cells in equation (13). There we denote , and the weights in the denominator satisfy to take into account different activation thresholds between naive and memory CD4⁺ T cells, similar to [14].

Activated CD4 + cells and activated memory CD4 + T cells can undergo several pathways: they can a) proliferate with probability q, or b) differentiate either to memory T cells or to effector T cells with probability respectively p or . The proliferation of activated T cells and occurs at maximum rates . In the case of transient immunogenicity we let , whereas .

To allow for immune memory effects we assume logistic growth in equation (12) for memory CD4 ⁺ T cells with growth rate

chosen to be equal to the maximum activation rate of memory CD4⁺ T cells by APCs. We note that other mathematical approaches are possible to maintain immune memory for longer periods of time, for example, in a model for the immune response to influenza [15].

In subsequent simulations we set p = 0.5 except for the case of restored immune tolerance (transient immunogenicity, where no memory T cells form).

We also include the antagonistic effects of cytokines in the activation rates. On the one hand, the anti-apoptotic action of TNF on CD4 + T helper cells [6,31–33] is modelled as a reduction in the respective removal rates by up to 50% at high concentration of TNF. Further, the increased activation rate due to pro-inflammatory TNF [33,34] is modelled as an increase of up to 50% of the maximum activation rates at high at high concentration of TNF. On the other hand, IL-10 has a suppressive effect on activation, proliferation and differentiation from naive to T helper cells [33–35], and this is modelled by a reduction of the maximum activation rates and the maximum proliferation rates by a factor in (11) and (13) as a proxy for the immunosuppressive action of IL-10, similar to equation (8).

Finally, MTX has a dose-dependent apoptotic effect on the activated T cells only (experimental evidence provided in [36]) which we model through a Hill function and .

(10)(11)(12)(13)(14)

with initial conditions

Observe from equations (11), (12), (13) that the parameters are not simultaneously identifiable.

2.5.4 Natural killer (NK) cells.

NK cells are characterised as CD3^-CD56⁺, and they produce cytokines like TNF, IFN-, which prime CD4⁺ T cells and stimulate maturation of APCs [37]. We denote by K₀ the total count of CD3^-CD56⁺ NK cells and by K₁ the mature cytotoxic NK cells under inflammatory conditions (identified as CD3^-CD56⁺CD16⁺ NK cells [38,39].). In homeostasis NK cells are removed at rate . TNF stimulates maturation of naive NK cells [40]. The maturation rate of NK cells increases as function of TNF in a dose-dependent manner.

Methotrexate has been reported to suppress NK cell activity in an in vivo murine model [41], downregulate gene expression in CD56 NK cells [42], as well as lower counts of NK cells and in human patients with rheumatoid arthritis treated with MTX compared to healthy subjects and MTX-naive patients [43]. Data from CoMARIS also indicate MTX leads to a reduction in NK cell counts, which we model as a suppressive effect on the two populations in equation (15). The dynamics of TNF-modulated activation of NK cells is described by

(15)

In the MTX– cohort we assume that the NK cell populations do not change between weeks and 0, so we set cells/mL, with K₁(0) = 60000 cells/mL, while in the MTX⁺ group we set cells/mL, with cells/mL.

If the two populations are in equilibrium at t = 0, the value of can be estimated by

2.5.5 B lymphocytes.

For the B lymphocyte dynamics we adapt the model in [14] to model the process of the T cell-dependent activation, and to keep the model structure simple and due to data availability we do not model explicitly the binding of antigen to B cell receptor as proposed by [14].

We take the value of the initial condition for (16) in the range of 1000–5000 cells/mL since initially a small number of naive CD19⁺CD27^- cells would be available for antigen-presentation by effector CD4⁺ T lymphocytes. We choose this cell type to represent naive B cells in the model, since the expression of the CD27 marker is presumably essential for the formation of memory B cells [44,45].

Naive and memory B cells are activated in a T cell-dependent manner by CD4 + T effector cells with a maximum rate . For the T cell-dependent activation of naive and memory B cells we make a similar assumption as we did before for the activation of CD4⁺ T cells. The activation rate depends on the relative abundance of effector T cells available due to the limited number of B cells to which an effector T cell can bind at any given time. We use saturation terms of the form proposed by [14] to model this assumption: for the activation of naive B cells in equation (16) and for the activation of memory B cells in equation (18). Here, the notation stands for the total amount of naive B cells, activated naive B cells, memory B cells and activated memory B cells:

The weights satisfy to take into account the fact that effector CD4⁺ T cells activate memory B cells more efficiently than naive B cells [14] for the ADA-high patient cohort. For the ADA-low patient cohort, those values are assumed to be similar in magnitude because of the assumed lower number of T memory cells overall.

The rates of removal of activated B cells is given by respectively in (17) and (19). Activated naive B cells become short-lived plasma cells at constant rate , whose value may depend on the specific patient cohort. We assume that activated memory B cells are not able to differentiate into short-lived plasma cells.

We introduce parameters to describe the fates of activated naive and memory B cells . These cells can a) proliferate with probability s or b) differentiate into memory B cells with probability r, or into long-lived antibody-secreting plasma cells P with probability . Activated B cells proliferate in a T cell-dependent manner at maximum rate , and activated B memory cells proliferate in a T cell-dependent manner at maximum rate . When , few B lymphocytes differentiate into B memory cells, and when , few activated B lymphocytes differentiate into long-lived plasma cells. In subsequent simulations we set r = 0.5 except for the case of restored immune tolerance (transient immunogenicity, where immune memory does not form in the form of B memory cells).

In the equation for memory B cells (18) we introduce a logistic term with growth rate to account for the maintenance of immune memory over long periods of time. Finally, MTX has a dose-dependent apoptotic effect on the activated B cells only (experimental evidence provided in [12]) which we model through linear rates and . The dynamics of B lymphocytes is described by

(16)(17)(18)(19)

with initial conditions

2.5.6 Short-lived plasma cells.

Short-lived plasma cells are primarily found in secondary lymphoid organs and serve as the initial producers of antibodies during an immune response. In contrast, long-lived plasma cells are generally located in the bone marrow, where they sustain antibody production for months to years. The role of short-lived plasma cells is important for the case of transient immunogenicity (observed in some ADA-low patients), whereby antibodies are formed initially, but whose production is not maintained over time. Short-lived plasma cells are formed by activated naive B cells and are removed at rate .

2.5.7 Long-lived plasma cells.

The long-lived plasma cells P are essential for the immune memory. They are able to produce antibodies over long periods of time. They are formed from activated naive and memory B cells with probability .

(20)

2.5.8 B10 cells.

B10 (or B regulatory, Breg) cells produce IL-10 and act as regulators of immune response [30]. They are known for actively secreting IL-10 and can influence the immune response in multiple ways: by suppressing TNF responses in vitro, IFN responses in vitro and in vivo, CD4⁺ T cell activation, dendritic cell antigen presentation and antigen-specific CD4⁺CD25⁺ proliferation [46]. Studies suggest that on one hand, MTX leads to increased interleukin-10 gene expression in lymphocytes in vitro [47] and, on the other, MTX-induced immune tolerance induces a specific expansion of IL-10- and TGF--secreting B cells that express Foxp3, which implies the induction of B regulatory cells [11,48]. CoMARIS data (Fig C in S1 Appendix) demonstrate that the abundance of CD24^highCD27 + cells, which are able to differentiate into B10 cells [49], is stimulated in the MTX+ cohort.

We use the following model to describe the stimulatory action of MTX on B10 cell population:

(21)

The mathematical model developed and presented here consists of a relatively large number of equations (23 in total), while only a few of the parameters involved can be directly found in the literature. In addition, many of these parameters are not simultaneously identifiable if one attempts to estimate them from available data. To overcome this limitation, we employ simpler, lower-dimensional auxiliary models—typically involving only two or three variables to estimate specific subsets of parameters. These models are presented in Section C in S1 Appendix. The resulting parameter values were then integrated back into the full model, ensuring consistency between the simplified and comprehensive formulations. The values of the model parameters are listed in Section D in S1 Appendix. These parameters are chosen to calibrate the model output representing virtual patients whose status is classified as ADA-low (including ADA-transient) and ADA-high according to the criteria from [9].

Our mathematical model involves a large number of parameters, and many of them have been estimated using simpler auxiliary models. Given the model’s complexity, it is essential to assess how variations in individual parameters influence the outputs of interest ADA titre A and adalimumab concentration U.

Sensitivity analysis is a mathematical technique used to determine how variations in model parameters affect model outputs. It helps identify which parameters have the greatest impact, guiding model refinement and interpretation. Sensitivity measures can be local, assessing small perturbations around a nominal parameter set, or global, evaluating parameter influence over a wider range. In our study, we focus on local sensitivity analysis to capture the immediate response of the system to small changes in parameters.

This approach allows us to quantify the dependence of model predictions on specific parameters, helping to identify key contributors to system dynamics and potential sources of uncertainty. By systematically evaluating parameter influence, we ensure the robustness of our findings and gain a deeper understanding of the model’s behaviour.

We perform local sensitivity analysis using a sensitivity matrix to understand the effect of changes of individual parameters on the model output [50]. This analysis is performed numerically in R with the FME toolbox [51].

For a parameter the element of the matrix , where is the computed value at day . Cumulative measures of the local sensitivity of the parameter are norms such as as well as the mean and extremes . This produces a ranking of the importance of the changes of individual parameters on the outputs of interest.

3.1 Local sensitivity analysis

The ADA titres reported in the data set suggest a finer stratification of the ADA-low and ADA-high cohorts, and we reflect this structure in the sensitivity analysis and construction of virtual cohorts. For the ADA-low patient cohort we perform three separate sensitivity analyses: for the subcohort with transient immunogenicity (3 patients), for the subcohort with ADA titre below 30AU/mL (10 patients), and for the subcohort with ADA titre between 30 and 100 AU/mL (3 patients). The parameters with the highest sensitivity index (>0.5 in absolute value) with respect to the adalimumab concentration U and ADA titre A from the subcohort with ADA titre below 30AU/mL are , for the subcohort with ADA titres between 30 and 100 AU/mL the results are similar (Fig 2A-2B). For the transient ADA subcohort the model parameters with largest sensitivity indices are (Fig 2C).

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Fig 2. Local sensitivity analysis (sensitivity indices): the subcohort in ADA-low group with ADA < 30 AU/mL (A); the subcohort in ADA-low group with ADA between 30 and 100 AU/mL (B); the subcohort with transient ADA (C).

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For the ADA-high patient cohort we perform two separate sensitivity analyses: for the subcohort with ADA titre below 1000AU/mL, and with ADA titre above 1000 AU/mL (observed in just 2 patients). For both ADA-high subcohorts the local sensitivity analysis shows the parameters with the highest sensitivity index (>0.5 in absolute value) with respect to the adalimumab concentration U and the ADA titre A are , , , p,q,r,s, (Fig 3).

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Fig 3. Local sensitivity analysis (sensitivity indices): the subcohort in ADA-high group with ADA < 1000 AU/mL (A); the subcohort in ADA-high group with ADA > 1000 AU/mL (B).

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3.2 Virtual patient cohorts

Since the available patient data from CoMARIS do not contain counts of APCs, antigen-specific T or B lymphocytes, and many parameter values have unknown values or are non-identifiable from the data, we employ Latin hypercube sampling for the parameters exhibiting largest sensitivity indices (>0.5 in absolute value) to generate virtual patient cohorts corresponding to ADA-low (including transient ADA) and ADA-high cohorts. In order to represent the structure of the virtual patient cohorts according to the titre ranges for the ADA-low, ADA-high patients and those with transient immunogenicity, we vary the parameters with highest sensitivity indices in a Latin hypercube sample (). The reported ranges of their values in Section D in S1 Appendix reflect selective differentiation of the virtual patients driven by the data ranges of ADA titre in weeks 4, 8, 12 and 26 following the criteria for ADA-low and ADA-high patients [9], and the respective adalimumab concentration range. As the parameter has lower sensitivity in the ADA-high cohort, it is assumed to be fixed there. The parameter has a higher sensitivity in the transient ADA subcohort, so it is assumed to vary only there in order to produce virtual patients that conform to the ranges of the data. The probabilities p,q,r,s are assumed fixed. The range for the Latin hypercube for the parameters representing the putative therapeutic effects of MTX is chosen as mean value 15% (denoted henceforth as LH15), similar ranges are used to generate virtual cohorts in [52,53]. Furthermore, the point value of the PK parameter which is varied by agrees with the range of obtained by fitting individual patient data of adalimumab, for whom the fitted value , in addition, agrees with the value reported elsewhere [54].

Pearson correlation coefficients for the parameter pairs with statistical significance at level 1% in the ADA-low, ADA-high and ADA-transient virtual patient cohorts are summarised in Table K in S1 Appendix. The observed correlations are generally moderate and do not systematically involve those parameters with the highest sensitivity indices (Figs 2 and 3). The high correlation between may result from the assumption reflecting the different activation thresholds between naive and memory CD4⁺ T cells.

The selected parameter values allow us to explore the effect of parameter variability on the dynamics of the model by computing the solution for each virtual patient. Some solutions for the model without co-administration of MTX are plotted in Figs 4-6. We see that the dynamics for the low and ADA-high cohorts corresponds quite satisfactory to the patient data on adalimumab concentration and ADA titres. In the ADA-low cohort due to the nature of data reporting (ADA-negative values are reported as <12 AU/mL), the plotted values for the ADA titre use this cut-off value, so the minima are at 12AU/mL. Hence, the plotted data range for week 26 in Fig 5, left panel, appears to majorise the simulated trajectories. Simulations of ADA and adalimumab dynamics over a period of 52 weeks are provided in Fig Y in S1 Appendix and reveal that despite the different temporal profiles the classification by cohorts remains valid for the prolonged period.

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Fig 4. Selected MTX– virtual patients from ADA-low cohort.

Error bars represent minimum and maximum (16 patients). Trajectories use parameter values listed in Section D in S1 Appendix. The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

https://doi.org/10.1371/journal.pcbi.1014740.g004

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Fig 5. Selected MTX– patients from virtual cohort with transient ADA.

Error bars represent minimum and maximum (3 patients). Trajectories use parameter values listed in Section D in S1 Appendix. The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

https://doi.org/10.1371/journal.pcbi.1014740.g005

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Fig 6. Selected MTX– patients from virtual ADA-high cohort.

Error bars represent minimum and maximum (7 patients). Trajectories use parameter values listed in Section D in S1 Appendix. The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

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To test for the effect of MTX on the immune response in these virtual patients we prepare a list of scenarios based on the admissible combinations of modes of action (Table 2). A check mark (✓) means that MTX is assumed to act on the specific cell population, and the terms describing MTX action corresponding to the particular scenario are included in the respective equation of the model. For example, in scenario 6 in Table 2, MTX is used only in the equations for APCs (9) and activated T and B lymphocytes (equations (11), (13), (17), (19)). Or, further, scenario 23 assumes that MTX acts only on APC apoptosis and Breg cell activation as described in equations (7) and (21), and omits the terms with MTX action in the other equations. In the respective scenario, we draw the parameter value in the functional responses for each hypothetical mechanism of action of MTX from a uniform distribution with ranges mean or with mean and SE derived from the auxiliary models listed in Section C in S1 Appendix. Then, using the virtual patient cohort, we perform simulations of all scenarios for the mechanism of MTX action. Illustrative results are shown in Figs 8-10 for scenario no. 28.

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Table 2. List of model scenarios for the action of MTX on different cell types.

https://doi.org/10.1371/journal.pcbi.1014740.t002

The local sensitivity analysis gives us some indications of the scenarios that are most likely to affect the dynamics of ADA production. For example, the parameter that represents the T cell activation by APCs has a high sensitivity index in the ADA-low and ADA-high cohorts, and in some scenarios its value is affected by the value of c₁₀, serving as a proxy for Breg production of anti-inflammatory cytokines such as IL-10 [11,29]. Similarly, the maximum proliferation rate of activated T cells , another parameter with high sensitivity index is also affected by c₁₀. Hence, we would guess that scenarios where MTX is taken to increase abundance of Breg cells (scenarios with no. 1, 3–5, 8, 10, etc. in Table 2) are more likely candidates. However, we stress that the model incorporates this particular mode of action in a phenomenological manner, as levels of IL-10 are not available in the CoMARIS data set, and the suppressive effect of IL-10 on APCs, T lymphocytes, etc. is hard to quantify against published data [28,35] due to the variety of assays used. In particular, the dynamics of activated T cells can be directly influenced by the MTX-induced apoptotic effect represented in other scenarios such as no. 2, 9, 13, 24, etc.

Using the parameter values drawn from the Latin hypercube LH15, we generate 190 virtual patients for the ADA-low cohort (30 of whom are transient ADA, 130 of whom have ADA titre below 30 AU/mL), and 100 for the ADA-high cohort (20 of whom have ADA titre above 1000 AU/mL). These cohort sizes are chosen as approximately 10-fold the cohort size reported in [9] and conform to the relative shares of the respective sub-cohorts according to ADA levels. We do not compute individual trajectories for 260 virtual patients for the ADA-negative cohort who do not receive MTX because those can be simulated by setting parameter values (no formation of Ag-specific T or B lymphocytes), (no maturation of APCs) or (no loading and processing of antigen).

To simulate the different scenarios of the mechanism of action of MTX on the immune cell populations in the model, we perform simulations by substituting the values X(t) resulting from the MTX pharmacokinetic equation (4) with mg/mL into the equation(s) describing the dynamics of the respective cell population for each virtual patient as outlined in Table 2.

Then the resulting values of antidrug antibody titres at week 4, 8, 12 and 26 are compared against the criteria outlined in [9] and the MTX+ virtual patient is assigned one of 3 status categories: ADA-positive MTX+ at week 26, ADA-high MTX + , or ADA-low MTX + . The category ADA-negative MTX + is complementary to the last two, and not included in the estimation. To carry out this analysis we derive the conditional probabilities for a given patient transitioning between cohorts under MTX administration, denoted by , for each cohort .

Based on the parameter values that characterise the ADA-negative MTX– patients, we assume that for any virtual patient from the ADA-negative MTX– cohort, the probability to become ADA-negative under co-medication of MTX, in any scenario. For any scenarios with index we compute the probability distribution for the 3 status categories

Due to our assumption on the conditional probabilities that

the above expressions simplify.

The probabilities from the experimental observations in CoMARIS [9] are shown in Table 3, and compared against the probability distributions computed across all 31 scenarios in Fig 7. Histograms of the conditional probabilities resulting from the virtual patient cohorts drawn from the Latin hypercube LH15 are shown in Fig Z in S1 Appendix.

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Table 3. Experimental observations of the probabilities based on [9]. p₁ presents the probability of ADA presence at week 26 only, whereas p₄ represents the probability of being ADA-negative across all visits.

https://doi.org/10.1371/journal.pcbi.1014740.t003

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Fig 7. Computed probabilities for the Latin hypercube LH15 with methotrexate dose mg/mL for the 31 scenarios (horizontal axis).

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Next, we define a metric to rank the hypothetical scenarios from most to least likely based on the model prediction. For scenario with no. we compute the squared residual deviation of the computed probabilities from the reported values (j = 1,2,3) (since ):

(22)

To account for differences in model complexity across scenarios, we introduce an AIC-inspired model selection criterion. Unlike the AIC, our criterion is not derived from a likelihood function but is specifically designed to compare mechanistic scenarios by balancing the discrepancy with the experimental data against the number of active biological mechanisms. Consequently, standard interpretations of values and associated thresholds are not directly applicable to the proposed criterion; a detailed mathematical interpretation of the criterion, including its relation to the underlying discrepancy measure, is provided in Section A in S1 Appendix. Since the structure is fixed in the basic model without MTX, we use the number of active mechanisms across scenarios as a proxy for model complexity. The resulting metric is defined as

(23)

where n = 3 is the number of summary statistics and is the number of active mechanisms in scenario i. The scenario with index is selected by the proposed criterion as the preferred mechanistic scenario within the present modelling framework.

The results in Table 4 show that scenario no. 28 achieves the lowest IC value. This scenario corresponds to a mechanism of MTX action inducing apoptosis of activated CD4 + T lymphocytes only. While several more complex scenarios (e.g., no. 13, 21, 25, 27, 29, and 31) yield comparable goodness of fit, their improved fit is marginal and does not compensate for the increased model complexity. In particular, scenarios no. 27 and no. 29 incorporate MTX effect respectively solely on APCs and NK cells, scenario no. 13 includes apoptosis of both activated B and T lymphocytes, and scenario no. 21 represents a combination of suppression of APC and activated T lymphocytes, while scenario no. 25 additionally incorporates activation of regulatory B cells. Overall, these results indicate that increased apoptosis of activated CD4 + T cells alone provides the most parsimonious explanation of the observed data within the current modelling framework, while additional mechanisms cannot be reliably identified given the available data. The resulting ADA profiles in the other scenarios do not match in a staisfactory manner the reported ADA ranges in MTX+ patients from CoMARIS data (Section E.2 in S1 Appendix).

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Table 4. Ranking of scenarios according to the AIC-inspired criterion (23), with mg/mL. Lower IC values indicate a better trade-off between goodness of fit and model complexity.

https://doi.org/10.1371/journal.pcbi.1014740.t004

Illustrations of simulated adalimumab and ADA dynamics for selected virtual patients under scenario no. 28 are plotted in Fig 8 (ADA-negative), Fig 9 (ADA-low), and Fig 10 (ADA-high). The profiles of adalimumab and ADA for the entire virtual patient cohort under this scenario are shown in Fig I in S1 Appendix. Section E.2 in S1 Appendix containes the profiles for several scenarios with lowest IC (no. 13, 21, 25, 27 and 29) as illustration. Cell population dynamics for these four scenarios are illustrated in Sections E.5-E.9 in S1 Appendix.

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Fig 8. Selected virtual patients from the MTX + ADA-negative cohort under scenario no. 28 (MTX effect on activated T lymphocytes only).

Error bars represent minimum and maximum of the CoMARIS data (35 patients). Trajectories use parameter values listed in Section D in S1 Appendix and drawn from LH15 with MTX concentration dose mg/mL.

https://doi.org/10.1371/journal.pcbi.1014740.g008

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Fig 9. Selected virtual patients from the MTX + ADA-low cohort under scenario no. 28 (MTX effect on activated T lymphocytes only).

Error bars represent minimum and maximum of the CoMARIS data (14 patients). Trajectories use parameter values listed in Section D in S1 Appendix and drawn from LH15 with MTX concentration dose mg/mL. The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

https://doi.org/10.1371/journal.pcbi.1014740.g009

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Fig 10. Selected virtual patients from the MTX + ADA-high cohort under scenario no. 28 (MTX effect on activated T lymphocytes only).

Error bars represent minimum and maximum of the CoMARIS data (3 patients). Trajectories use parameter values listed in Section D in S1 Appendix and drawn from LH15 with MTX concentration dose mg/mL.

https://doi.org/10.1371/journal.pcbi.1014740.g010

3.3 Simulation of reactive co-administration of MTX

We employ the model to simulate a setting of reactive co-administration of MTX. That involves monitoring of adalimumab concentration and ADA titre for the patients and introduction of MTX if ADA are detected in the patient blood sample.

We run the model under scenarios no. 28, 13, 24, and 29, which have the lowest values of the (Table 4) and introduce a delay in MTX co-administration ( mg/mL) in weeks 4, 8, or 12 (Figs 11, 13, and S-X in S1 Appendix), and simulate the dynamics over the following weeks.

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Fig 11. Simulation of the temporal dynamics for ADA titre and adalimumab concentrations for virtual patients from the ADA-low (A) and ADA-high (B) cohorts under scenario 28, with co-administration of MTX starting in week 4.

The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

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Under scenario no. 28, which our analysis suggested as the most likely mechanism whereby MTX acts on immune cell populations, the results show that if MTX co-administration begins in week 4, some ADA-low virtual patients remain below the ADA titre threshold of 12 AU/mL, or have a rapid reduction in ADA titre following the MTX co-administration, whereby the adalimumab levels (Fig 11A) are similar to those in the CoMARIS MTX⁺ cohort [9]. Thus, on the one hand, for an ADA-low patient the decrease in adalimumab concentration is already latent and MTX maintains a concentration that tends to increase after 7 weeks. In the ADA-high virtual patient cohort, on the other hand, the group splits into two: some that maintain very high ADA titre (>200 AU/mL) and some that have a lower ADA titre, and adalimumab levels maintained very low or close to half of those in the ADA-low cohort respectively (Fig 11B). Thus, for some ADA-high patients delayed co-administration of MTX results in a partial restoration of adalimumab levelss, but for others, there is no significant impact on adalimumab concentration compared to the case when MTX is administered from week , before the TNF inhibitor therapy begins.

On the contrary, if MTX co-administration begins in weeks 8 or 12, there is a decrease in the concentration of adalimumab compared to the case where MTX is introduced from week in the ADA-high cohort. If the patient is classified as ADA-low, then the dynamics resembles more closely the one that observed in the CoMARIS data set, and the levels of adalimumab is eventually restored towards week 20 and beyond if MTX co-administration starts in week 8 or 12 (Figs 12A and 13A). However, if the patient is classified as ADA-high and has ADA titre > 1000 AU/mL at week 12, then MTX co-administration does not bring any restoration of adalimumab levels, which remain below 2–3 g/mL (Figs 12B and 13B). This difference may be explained by the time lag that is required for the adaptive immune response (emergence of adalimumab-specific T and B lymphocytes and plasma cells). If the MTX co-administration begins after the start of the adaptive immune response, for the ADA-high patients it would cause no effect in counteracting immunogenicity. Results for delayed MTX co-administration in the scenarios no. 13, 24 and 29 are similar and shown in Section E.8 in S1 Appendix.

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Fig 12. Simulation of the temporal dynamics for ADA titre and adalimumab concentrations for virtual patients from the ADA-low (A) and ADA-high (B) cohorts under scenario 28, with co-administration of MTX starting in week 8.

The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

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Fig 13. Simulation of the temporal dynamics for ADA titre and adalimumab concentrations for virtual patients from the ADA-low (A) and ADA-high (B) cohorts under scenario 28, with co-administration of MTX starting in week 12.

The LOD of the antibody assay (12 AU/mL) is indicated as a dotted line.

https://doi.org/10.1371/journal.pcbi.1014740.g013