Mathematical model

To better understand MV-BiTE therapy, we generated a system of ordinary differential equations (ODEs) based on previous models for the immune response to immunostimulatory oncolytic viruses [47,48]. Our model captures a growing uninfected population of tumour cells, U(t), that become infected by MV-BiTEs, V(t) and lyse. Inside infected tumour cells, the virus replicates, leading to the production of BiTEs, B(t), which are released upon cell lysis. The presence of dead cells or cell debris D(t) stimulates the recruitment of effector CD3⁺ T cells T(t) to the tumour site. T cells can then induce apoptosis in tumour cells. The presence of BiTEs increases the ability of effector T cells to induce apoptosis in tumour cells. These dynamics are captured in Fig 2, Tables 1 and 2 and the system of ODEs below:

(1)

(2)

(3)

(4)

(5)

where t is time and N = U + D + T. To keep the units consistent throughout the model, V is in units cells which is converted to a quantity of virions by , where has units virions/cell and has units virions. For further examples of mathematical models that capture the immune response to oncolytic virotherapy and the resulting antitumour effect, see [28,34].

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Table 1. Summary of variables in Eqs. 1–5.

https://doi.org/10.1371/journal.pcbi.1014533.t001

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Table 2. Parameters for Eqs. 1–5.

Estimates were obtained through fitting to in vitro and in vivo data with parameters having different values in vitro (top row) and in vivo (bottom row). Other measurements were estimated based on literature, with their reference given. For the annotation of the experiments below they correspond to the following experiments detailed in the methods: “progeny” - Viral replication kinetics (S1 Fig), “viability” - Virus-mediated (onco-)lysis:cell viability (S1 Fig), “lysis” – BiTE-mediated (T cell) cytotoxicity: specific lysis (S2 Fig), “tumour volume” – in vivo tumour volume (Fig 4).

https://doi.org/10.1371/journal.pcbi.1014533.t002

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Fig 2. Schematic for the mathematical model in Eqs. 1–5 describing MV-BiTEs.

Viruses V(t), genetically engineered to encode BiTEs, are injected into the system and infect uninfected tumour cells U(t). These cells eventually die, becoming dead cells, D(t) and release new viruses and BiTEs B(t) to the tumour environment. The presence of dead cells activates and recruits CD3⁺ T cells T(t). T cells induce apoptosis in cancer cells. The presence of BiTEs increases the ability of T cells to induce apoptosis in uninfected tumour cells. Created in BioRender. Engeland, C. (2026). https://BioRender.com/ia6sdw3.

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In Eq. 1, tumour cells, U(t), grow under a Gompertzian tumour growth law [49], under which cells proliferate at rate r and the population will grow until a certain carrying capacity L is reached. The carrying capacity is a representative of the tumour growth trajectory and does not represent the termination criteria, which may be reached at a smaller value. Tumour cells become infected and undergo lysis at a rate . These infected cells can form syncytia, which we have chosen not to explicitly model here. Infection is modelled as proportional to the concentration of virus and a Michaelis-Menten function with the concentration giving 50% of the maximum infectivity, also known as the half-effect. We introduce this infection term to capture that virus penetration into a tumour is often inhibited spatially either due to the microenvironment or otherwise [50]; see [35,41,51] for similar uses of this infectivity formulation. Tumour cells undergo T cell-induced apoptosis at a frequency-dependent rate with constant k. For uses of frequency-dependent terms in OV modelling similar to this, see [47,48,52]. To model the effect of BiTEs on T cell activity, we assume a baseline T cell apoptosis factor , and that BiTE-mediated killing is an additional factor on top of this baseline killing rate, modelled using a Michaelis-Menten term with half-effect .

We have removed the intermediate ‘infected’ population that is often present in models of viral infections [28,34,47,53,54] and instead have direct generation of virus and BiTEs through infection of tumour cells, similar to [55], so that production of BiTEs is modelled continuously over time and not in discrete bursts. We found that introducing the infected cell pool and its specific lysis rate introduced both structural and practical identifiability issues into the model. We also assume infected cells do not replicate after infection, meaning the process of infection is terminal for a cell. In this way, we capture the process of infection, lysis and immune cell recruitment through one population D(t).

In Eq. 2, we track the accumulation of dead cells or dead cell debris, D(t), through the induction of lysis by an OV or apoptosis by T cells. To promote the practical identifiability of our model, we chose not to explicitly capture the production of tumour or viral antigens, subsequent activation of antigen-presenting cells, cytokine/chemokine production and then T cell recruitment. As such, we use the accumulation of dead cells and debris as a proxy for this process. Dead cell debris is produced at the same rate that tumour cells die and is removed (decays) from the system at a rate .

In Eq. 3, new viruses are produced through virus infection of tumour cells. These are assumed to decay due to factors such as adaptive immune cell clearance, complement clearance, and sequestration, for example, by the reticuloendothelial system at a rate . As each cell produces virions, we scale the rate of production and decay of virions by to obtain an estimate for the number of virions, .

In Eq. 4, through cell lysis, new BiTEs, B(t), are released. These BiTEs decay at a rate , i.e., a pharmacokinetic parameter. We assume that these BiTEs consist of both T cell-bound and free BiTEs in the local tumour microenvironment (TME). While BiTEs can be secreted throughout the lifetime of the cell, we found separating the number of BiTEs released at lysis from the amount secreted up to lysis was not structurally identifiable with the given experimental measurements. As such, we assume that any BiTEs released up to lysis are included in the amount secreted at lysis .

In Eq. 5, T cells, T(t), accumulate at the tumour site based on the presence of dead cell debris at a rate s. These cells also decay at a rate . We assume T cells are recruited at a rate sD, which can be thought of as a proxy for antigen-presenting cell activation by viral antigens or tumour antigens, downstream stimulation of T cells in the lymph nodes, and tumour-directed T cell chemotaxis.

The model was simulated in MATLAB and solved numerically using ode45. Initial conditions are given below:

The model was solved piecewise to simulate n injections of BiTEs B₀ and virus V₀ at time :

for I = 1,...,n.

Structural and practical identifiability

Identifiability is an important component of mathematical modelling as it determines the capacity of a model to extract parameter estimates from data [56–59]. This is particularly important in biology, where parameterised models are used to make predictions about living systems [60,61]. We refined the model presented in Eqs. 1–5 numerous times to minimise identifiability issues. The model was found to be structurally identifiable using the open-source software DAISY [62,63]. Structural identifiability implies practical identifiability only for infinite data [56]. For finite data samples, there are numerous ways to evaluate practical identifiability [64–66]; for our dataset and model, we have analysed practical identifiability with a global and local sensitivity analysis.

Details of experimental and clinical measurements

To estimate parameters in Eqs. 1–5, we use a range of in vitro and in vivo experiments from oncolytic measles virus (MV) encoding and not encoding BiTEs [13], as well as a Phase II clinical trial for talimogene laherparepvec (T-VEC) [67]. Vaccine-derived, live attenuated MV vectors were designed based on established reverse genetics systems and were generated to encode BiTEs targeting murine CD3 and human CD20 [68]. For in vitro studies and the syngeneic subcutaneous tumour model in C57Bl/6J mice, murine melanoma-derived B16-CD20-CD46 cells stably expressing the BiTE target antigen human CD20 and human CD46 as entry receptor for oncolytic measles virus were used.

Parameter estimation

To estimate parameters in the model, a series of hierarchical parameter fits were conducted to in vitro and in vivo measurements using MATLAB’s inbuilt non-linear least squares optimisation algorithm lsqnonlin and the inbuilt multistart algorithm multistart. The viral infection and replication parameters and were estimated using the in vitro viral progeny and cell viability experiments, see in S1 Text and S1 Fig. The BiTE and T cell killing parameters , and were obtained through fitting the specific lysis (%) data, S2 Fig. The in vivo tumour growth rate without treatment, r and L, were obtained through fitting to the control data set, assuming that 10⁶ cells were injected at day 0 and cells [71].

Given the initial parameter estimates from the in vitro experiments, we then recalibrated parameters to the in vivo measurements. We simultaneously fit the model to the purified BiTEs, MV and MV-BiTEs measurements to obtain , and s. The decay rate of T cells was taken from data fitting, /day, [48]. The time from initiation of cell apoptosis to completion can occur as quickly as 2–3 hours. If we assume that debris can remain for another 2 hours after this, this gives a dead cell clearance rate of = 4.8/day [72]. For fitting the in vivo data: g, and V(0)=10⁶ cells. Further details of each fit are provided in the Supplementary Material.

Sensitivity analysis

We evaluated the sensitivity of our model predictions to perturbations to the obtained parameter estimates through a local and global sensitivity analysis. We conducted a pairwise local sensitivity analysis around the fitted parameter values, see Supplementary Material and S3 Fig. We also conducted a global sensitivity analysis by running a Latin Hypercube Sampling and calculating the Pearson’s Correlation Coefficient (PCC) for each parameter with the final tumour size. Other sensitivity analysis methods, such as eFAST, MeFAST or Sobol, would also have been appropriate here [73].

In silico clinical trial

Virtual clinical trials (VCT), also known as virtual patient cohorts, phase i trials, digital twins, and in silico clinical trials attempt to integrate the heterogeneity of actual patient responses and preclinical studies through a cohort of virtual patients or individuals [33,38–44,74].

Virtual patient generation from in vivo data.

There are many ways to generate virtual patients [41,45,46] and in this work, we created virtual cohorts using two different methods employed by previous works [41,44,45]. We first created a cohort of patients that matched individual in vivo mouse tumour growth trajectories using acceptance-rejection sampling (Fig 3: Method (1)). We sampled 250,000,000 parameter sets from uniform distributions centered around the fitted parameter values with bounds [0.001, 5]. We took the 400 parameter sets that gave the minimum residual to the individual tumour volume measurements, labelled “m1” to “m8”. The second method we used to generate a virtual cohort, was a similar approach to that developed by Kim et al. [44]. We converted the change from baseline measurements for the clinical trial data to a probability density function (PDF) for the final tumour size x (Fig 3: Method (2)). We then sampled 400 patients from this PDF and optimised the set of parameters to obtain an estimate for the final model size that matched the sampled patient using a genetic algorithm (MATLAB’s ga). This gave us a distribution of final tumour sizes and corresponding parameter values that matched the PDF obtained from the clinical data.

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Fig 3. Illustration of the two virtual clinical trial generation methods: Method (1) individualised growth cohort and Method (2) tumour burden matching.

Method (1) investigates heterogeneity arising in the in vivo measurements by matching the model to individual mouse tumour growth trajectories. Parameters are sampled from uniform multi-dimensional parameter distributions, and an acceptance-rejection algorithm is implemented, accepting samples that align closest with the data. Once a cohort of N samples for each individual tumour growth trajectory have been sampled, the distribution for these parameters, e.g., p₁, can be compared across the different trajectories. Method (2) generates the virtual cohort through converting measurements from a clinical trial measuring the change from baseline (CFB) to a probability density function (PDF) f(x) for the final tumour size x. From this PDF, N individual final tumour sizes are sampled and a genetic algorithm is used to determine the optimal parameter combination for the model to match the CFB (%) data.

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Parameterisation and sensitivity of a mathematical representation for MV-BiTEs

To calibrate the model, we fit parameters from the system of ODEs in Eqs. 1–5 to in vitro measurements for viral progeny and cell viability (S1 Fig), and specific lysis (S2 Fig). We assumed the viral infectivity and CD3⁺ T cell killing rate may vary in vivo and refit these using the in vivo tumour volume experiments (Fig 4A – 4E). Estimates for parameters can be found in Table 2 with more details in the Materials and Methods and Supplementary Information S1 Text. Qualitatively, we observed good agreement between the data and the model, supported by goodness of fit statistics, Table 3. To investigate the sensitivity of the parameterised model, we conducted a local (S3 Fig) and global (Fig 4F) parameter sensitivity analysis. The global sensitivity analysis suggested the immune-related parameters , and s had a negative correlation with the tumour size.

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Fig 4. Parameterisation of the model for MV-BiTEs using in vitro and in vivo experiments.

Generation of data is described in the Methods. (A) Experimental protocol for in vivo experiments with BiTEs, MV and MV-BiTEs. (B)-(E) Tumour volume in vivo in (B) control (mock) and (C) BiTEs only, (D) MV only and (E) MV-BiTEs treatment groups. Black circles with error bars are representative of the mean and standard deviation of the data and grey circles are individual data points. Solid blue lines represent the model’s approximation to the data. (F) Pearson Correlation Coefficient (PCC) for a Latin Hypercube Sampling on [0.01, 2] capturing the sensitivity of the estimated parameters.

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Table 3. Goodness of fit statistics for model fit to the data in Fig 4, with parameter values in Table 2.

https://doi.org/10.1371/journal.pcbi.1014533.t003

Capturing inherent in vivo variability in tumour responses

In the in vivo experiment, mice exhibited heterogeneous responses to MV-BiTE treatment, with some mice experiencing tumour progression and others experiencing regression (Fig 5). To examine the model’s ability to capture this heterogeneity, we sampled a virtual cohort using Method (1) (Fig 3) for each individual mouse, m1-m8. We obtained 400 parameter sets for each m1-m8 trajectory that gave the smallest residual to the data (Fig 5A – 5I, with example trajectories in S4 Fig). The model was able to capture the heterogeneous range of tumour growths exhibited by the mice upon MV-BiTE treatment.

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Fig 5. Obtaining model trajectories that match the tumour volume experimental measurements under the treatment schedule in (A).

In the experiments, the tumour volume of 8 mice was tracked after treatment with MV-BiTEs with m1 to m8 in (B)-(I). The measurements for the mice are plotted on the same axis and represented by black dots, while the trajectories of the 400 parameter values sampled that gave the lowest residual error are plotted as coloured lines. Zoomed in inserts are provided when the trajectories are hard to visualise on the fixed axis. (J) The mean and standard deviation of the relative value for the parameters for each m1 to m8 are given as an error bar plot. These parameter ranges have been scaled relative to the maximum so they can be seen on the same figure. The corresponding parameter values and pair-wise correlations can be seen in the S4 and S6 Figs.

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Variability between the tumour growths is captured by the relative mean and standard deviation for each parameter (Fig 5J, with full distributions in S5 Fig). Noticeable differences in the mean for exist, suggesting that virus administration is largely driving the heterogeneous responses in the mice. Furthermore, changes in and seem to correlate with either tumour growth (m2, m3, and m7) or tumour decline (m1, m4-m6). This suggests that tumour growth or decline could be linked to the efficacy of CD3⁺ T cells and the pharmacokinetics of BiTEs.

Examining correlations between parameters, we found that all individual growths, except m2, had a positive correlation between viral characteristics, and (S6 Fig), i.e., increasing one of these parameters resulted in an increase of the other. Furthermore, m2 - m6 all had a negative correlation between k and , i.e., increases in one parameter observed a decrease in the other. This suggests a link between viral parameters and , immune parameters k and , and the response to MV-BiTEs.

Clustering individual responses into growth, delayed growth and remission

The responses to MV-BiTEs treatment can be grouped into three cohorts: 1. regression (m1, m4, m5, m6), 2. growth (m2, m3, m7,) and 3. delayed growth (m8). We generated data for these three virtual cohorts by sampling 400 parameter sets equally split amongst the mice belonging to that cohort (Fig 6A – 6C). Clear separation between the response cohorts can be seen in the final tumour size on day 48 (the final measurement across m1-m8) (Fig 6D). Not all trajectories in the growth cohort (cohort 2) are expected to surpass the delayed growth cohort, with some exhibiting small tumour sizes at day 48. This suggests that while the delayed growth cohort may at first seem to fare better compared to the growth cohort, some favourable trajectories are expected in the growth cohort.

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Fig 6. Analysing cohort characteristics.

From the mice in Fig 5, three sub-cohorts were generated based on the dynamics in the data: regression (cohort 1), growth (cohort 2), delayed growth (cohort 3). (A-C) Mean tumour volumes and standard deviations for the three cohort values. (D) Final tumour cell number (as calculated for day 48) for each cohort. (E-F) For the three cohorts, the patterns within the parameter values are presented using scatter plots for vs k and vs . Points are coloured based on the cohort. (G-I) Histograms for the parameters , and .

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Examining correlations between model parameters across the cohort we see distinctions in the clustering patterns for k and (Fig 6E). This suggests that the relationship between the killing rate and the basal killing efficacy are driving the response to MV-BiTE: low basal killing efficacy () drives the growth cohort (cohort 2) and higher ratios of basal killing efficacy, and killing rates, drive the recovery cohort (cohort 1). Interestingly, there seems to be a bimodal relationship between these two parameters for the delayed growth cohort (cohort 3) suggesting that there are two possible explanations for tumour growth in this cohort. Similarly, there is clustering in the relationship between and (Fig 6F), whereby for small basal killing efficacy and values of BiTE clearance, , between 2 and 7 gives rise to the growth cohort (cohort 2). Comparing single parameter distributions (Fig 6G – 6I) the regression cohort (cohort 1) has very low and and the growth cohort (cohort 2) has very low but slightly higher , suggesting again that BiTE clearance is a key determinant of the response to MV-BiTEs.

Efficacy of administration protocol on variable tumour growths

For the three response cohorts (regression, growth, and delayed growth), we next investigated the effect of varying the MV-BiTEs administration protocol. We varied the number of dosages, n, and days between dosages, (Fig 7A) with the total virus dose kept constant. Variations to the MV-BiTEs protocol had a negligible effect on the regression cohort (cohort 1) (Fig 7B (top row) and S7 Fig), suggesting this cohort will always respond to MV-BiTEs irrespective of the administration protocol. A larger dependence on the protocol is seen for the growth cohort (cohort 2) and delayed growth cohort (cohort 3). The optimal protocol for these cohorts is more injections given further apart.

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Fig 7. The impact of the treatment protocol on the three cohort responses: 1) regression, 2) growth, and 3) delayed growth.

(A) Treatment protocols considered varied the number of injections n and days between injections . (B) The effect of the protocol on the mean tumour size (in mm3) for each cohort at either 48 days (top row) or 7 days after the final injection (bottom row). (C-D) Parameter correlations for cohort 2: (C) k vs and (D) k vs . (E) Parameter correlations for cohort 3 for k vs . All scatter plots are based on 7 injections given 6 days apart with parameter pairs represented as points coloured by the log tumour size at 48 days.

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Taking the optimal protocol to be 7 injections 6 days apart, we found that there is clear separation between the best and worst responses in the growth (cohort 2) and delayed growth (cohort 3) cohorts. For instance, in cohort 2, there is a clear separation in the pair-wise relationships between and , and and k, respectively (Fig 7C – 7D). This suggests that we can determine individuals who will respond best to this treatment protocol based on their BiTE pharmacokinetics, viral infectivity and CD3 killing rates. Similarly, in the delayed growth cohort (cohort 3) (Fig 7E) individuals who respond best have high CD3 killing and basal kill rates. We then examined whether this optimal protocol holds under the condition that after treatment ends, we resect the tumour 7 days later (Fig 8B (bottom row)). In this scenario, all protocols are similar as 7 days after treatment, the model predicts all cohorts are responding to MV-BiTEs. The rationale behind this metric is that it helps to evaluate the efficacy of treatment in controlling tumour growth by examining the impact once the treatment is halted and the tumour has time to regrow.

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Fig 8. Expanding a virtual patient cohort informed by the Phase II T-VEC clinical trial [67].

(A) Change from baseline (%) for the Phase II T-VEC clinical trial digitized using WebPlotDigitiser [76]. (B) Distribution of initial tumour diameters for 400 virtual patients sampled from a gamma distribution estimated from the summary statistics of Dessinioti et al. [75] overlaid as vertical lines: 10.6% (dashed orange), 25% and 75% (blue dotted), and median (black dot dash). (C) Violin plots comparing the change from baseline for the clinical trial (CT) measurements and model simulations from the sampled initial tumour diameters (IC). (D) Scatter plot for parameters r vs k obtained from optimising the model output to the CT measurements using a genetic algorithm, i.e., Method (2). Each disk represents a patient with the colour representing the tumour cell count. (E) Plot showing the agreement between the model (solid lines) and patient samples (open circles) for 7 virtual patients (colour matched). (F) The distribution for the total number of tumour cells predicted by the model at T = 23 weeks and those in the CT, the samples for the 400 virtual patients (VCT), and model predictions using a genetic algorithm (GA).(G) Summary of the original clinical trial dosage protocol and the 10 hypothetical protocols considered (Table 4). Coloured bars represent the strength of a dosage given in a particular week. (H) The final tumour cell counts for varying treatment protocols examined (s1-s5, e1-e5) compared to the original protocol 0. (I) Kaplan-Meier survival curves corresponding to (E) for a threshold of a 120% CFB. Note the survival probability range is plotted on [0.15 to 0.3].

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Case study: Virtual cohort representative of human responses

To examine the robustness of our model and its predictions for a clinical setting, we generated a virtual cohort using the Phase II clinical trial of T-VEC [67]. As no in-human data of MV-BiTEs therapy exists, we used a study of an advanced oncolytic virus encoding an immunostimulatory agent as a surrogate, here a herpes simplex type 1-derived OV encoding GM-CSF, as a surrogate. Melanoma was the targeted tumour entity in both the pre-clinical MV-BiTEs study and the T-VEC trial. In this clinical trial, an intratumoural injection of was administered initially, followed by fortnightly booster injections of starting 3 weeks after the initial T-VEC injection. The percentage change from baseline (CFB) for 98 patients was then recorded (Fig 8A). We assumed the same dosage size for MV-BiTEs as used for T-VEC in the clinical trial and included the addition of BiTEs in the injection to match our in vivo experiments. We fixed the final time of the simulation as T = 23 weeks (T = 161 days) and measured the resulting tumour size U(T)=U(161).

We first examined whether heterogeneity in the initial tumour burden could explain the change from baseline (CFB) measurements in the clinical trial. To do this, we parameterised a gamma distribution using summary statistics reported by Dessinioti et al. [75] for the diameter of melanomas in 537 cases (see Methods and Supplementary Information for details). We then sampled 400 virtual patient initial tumour diameters (Fig 8B) and converted these to initial cell counts to feed into the model. The resulting CFB predicted from the model for heterogeneous initial tumour sizes was unable to capture the range of the clinical trial measurements (Fig 8C). This suggests that changes to parameters, representing patient-specific characteristics, are needed to capture clinical outcomes.

We next chose to follow Method (2), (Fig 3) to generate a cohort of 400 virtual patients (Method (2), Fig 3). We sampled 400 CFB to create a larger sample of human patients and ran a genetic algorithm to determine optimal parameter sets that matched the sampled CFB (S8 Fig). We fixed the initial tumour size to the median melanoma diameter and chose three parameters to optimise, , and r (Fig 8D), based on their sensitivity so far revealed in our analysis. The genetic algorithm returned a close match between individual virtual patient samples and the model (Fig 8E – 8F). The distributions for the change from baseline of the real patient cohort and virtual patient cohort qualitatively match and passed the Kolmogorov-Smirnov test for similarity. We then examined the correlations between patient parameters and their resulting change from baseline (Fig 8D, with all correlations in S8 Fig). We found a particularly strong correlation between the tumour growth rate, r, and the T cell killing rate, k, with low growth rates and strong immune responses showing the largest negative CFB. The clearance of BiTEs, , also seems to positively correlate with CFB suggesting that a faster clearance of BiTEs results in a larger tumour size.

Case study: Impact of alternate treatment protocols on in silico human clinical trial

Lastly, we chose to examine whether an alternate, more effective, protocol exists. We generated 10 hypothetical treatment protocols that examined either the standard booster onset time (i.e., 3 weeks following the first dose) or an earlier booster onset time (i.e., 1 week following the first dose) and then varying intervals between doses and their multiplicity (Table 4 and Fig 8G). The total administered dose of MV-BiTEs, i.e., was fixed across all protocols considered. Reduction in tumour size was largest under the standard booster onset (3 weeks after the first injection) with more injections given more regularly (s4 and s5, Fig 8H). In contrast, intervals between consecutive doses larger than the current standard fortnightly protocol, do not improve the average tumour burden (i.e., s1-s3, e1-e3). To examine the relationship between final tumour size and hypothetical survival curves for our virtual cohort, we have plotted the Kaplan-Meier survival curves under the 10 protocols for a survival threshold of 120% CFB (Fig 8I). These suggest that individuals predicted to have large tumour sizes early on under the standard protocol will benefit in the short-term from the early booster onset protocols with frequent smaller dosages (e4 and e5).

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Table 4. Summary of the standard clinical trial protocol and the 10 hypothetical protocols examined in Fig 8. Note that the frequency and total doses are given from the booster dose onwards.

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