Biological system analysis

Abstract

Methods and apparatus for generating values for parameters of a model defining a relationship between one or more biological processes of a biological system. A plurality of data sets are received, ea

Description

[0001]Biological System Analysis Many biological systems are highly complex, with multiple different processes interacting with one another. Understanding such biological systems and the interaction between different processes is useful in a large number of applications such as the development of therapies. Quantitative systems pharmacology has been proposed for and used to model biological systems in the development of some therapies. Quantitative systems pharmacology uses models derived from experimental data to model biological mechanisms and processes such as drug mechanisms and disease processes. For example, immunotherapy is a type of treatment that uses an immune system of a patient to treat a condition. Immunotherapies have been identified as having potential to revolutionise treatment for cancer patients. Immunotherapies can, for example, provide treatments having fewer side effects than traditional treatments, and providing long term protection against the disease. The potential for immunotherapies has lead in recent years to an increase in development of new immuno-oncological therapies. Developing successful immunotherapies is difficult due to their complex systems of action and a lack of effective tools to predict success of treatments. This leads to challenges in developing models to design, build and test immunotherapies effectively. While quantitative systems pharmacology models have shown promise for developing immunotherapies, these models are complex to develop, require large amounts of data from experiments and the models can still remain challenging for use in development of immunotherapies. Other complex biological systems such as those governing the interaction of expressed genes and / or quantities of gene products in an in vivo or in vitro model, for example under the influence of a therapy, and biological systems used for bioproduction in which complex organic materials are produced under particular manufacturing conditions face similar challenges to immunotherapy. According to at least one aspect there is provided a computer-implemented method of generating values for parameters of a model defining a relationship between one or more biological processes of a biological system, the method comprising: receiving a plurality of data sets, each data set associated with a property of the biological system under a respective set of experimental conditions of a plurality of sets of experimental conditions, each data set associating a respective output value for the property of the biological system with the respective experimental conditions; receiving a model defining a relationship between the one or more biological processes of the biological system, the model comprising a plurality of model components, each model component of the plurality of model components defining a respective relationship between a plurality of model parameters, wherein a model component of the plurality of model components defines a relationship between the plurality of model parameters and the property of the biological system; for each model parameter of the plurality of model parameters, determining a plurality of initial model parameter values, each value of the plurality of initial model parameter values associated with a respective set of experimental conditions and determined based on a respective data set of the plurality of data sets; and for each model parameter of the plurality of model parameters, processing the plurality of initial model parameter values and the respective set of experimental conditions to generate updated model parameter values. Given the complexity of many biological systems, obtaining experimental data, whether by carefully configured real world experiments or from computational models, can be complex. Experimental data can be inaccurate and difficult to interpret. While attempts have been made to model biological systems based on experimental data, doing so effectively is challenging, with insufficient data available in many cases, and models overfitting even when additional data is available. By first determining a plurality of initial parameter values for model parameters based on a plurality of experimental data sets, and subsequently generating updated model parameter values based upon the initial parameter values, it has been found that information from the plurality of experimental data sets can effectively be combined to provide improved fitting of the model. While different sets of experimental data provide independent experimental results, the methods described herein can effectively extract interrelationships between the sets of experimental data to improve model fitting. The methods described herein use a hybrid approach in which standard model fitting techniques are applied, and then enhanced using machine learning techniques. Processing the plurality of model parameter values and the respective set of experimental conditions to generate updated values for the plurality of model parameters may comprise training a regression model based on the plurality of model parameter values and the respective set of experimental conditions. This processing step differs from conventional training techniques using Artificial Intelligence because the regression model is trained on the plurality of model parameter values and the respective set of experimental conditions derived from the experimental data rather than the data points of the experimental data. The regression model may be a Bayesian regression model such as a Gaussian Process model. Processing the plurality of model parameter values and the respective set of experimental conditions to generate updated values for the plurality of model parameters may comprise: generating one or more bounds for the values of the model parameter. An updated plurality of model parameter values may be determined based on the respective data set of the plurality of data sets and the one or more bounds. The one or more bounds for the values of the model parameter may comprise a range of values for the model parameter. The range of values for the model parameter may comprise an average value and a standard deviation. Determining an updated plurality of model parameter values based on the respective data set of the plurality of data sets and the one or more bounds may comprise: for each model parameter of the plurality of model parameters, determining a plurality of updated model parameter values, each value of the plurality of updated model parameter values associated with a respective set of experimental conditions and determined based on a respective data set of the plurality of data sets. The determining of an updated plurality of model parameters may therefore be bounded in a second and each subsequent iteration. The bounds are effectively extracted from the set of model parameter values as a whole, and can then be used to improve standard model fitting techniques with additional information that would not normally be available. The method may further comprise fitting the model to a respective data set of the plurality of data sets to determine a respective initial model parameter value of the plurality of initial model parameter values for each model parameter of the plurality of model parameters. Fitting the model to a respective data set of the plurality of data sets may comprise minimising a cost function associated with the model component of the plurality of model components that defines a relationship between the plurality of model parameters and the property of the biological system. The plurality of data sets may be obtained based upon experimental data. The plurality of data sets may be obtained directly from experiments under the sets of experimental conditions or obtained from computational models that are based on real world experimental data. In either case, the plurality of data sets are therefore based upon real world measurements derived from experiments. In some embodiments, data sets from different computational models can be mapped to a common model such that methods can additionally be used to determine a relationship between different known models, and can be used to combine existing models and data sets from the literature to provide improved accuracy and interpretability of existing models. The number of model parameters may be smaller than the number of experimental conditions. The model may therefore provide a simplified model of the real experimental conditions that are more readily interpretable, and additionally less prone to error. The model may define a relationship between expression of one or more genes and one or more biological processes of a biological system. The model may define a relationship between biological processes of an immuno-oncological system. The model may define a relationship between the bioproduction of biomolecules and one or more biological processes in a cell line. The model comprises one or more components each related to each other by one or more parameters. The components may relate to, for example, one or more of tumour size, cell size, or quantity of a biomolecule such as a gene, RNA or protein. The parameters may comprise, for example one or more cell growth rates, cell proliferation rates, cell death rates, transcription rates, translation rates, cell degradation rates, and binding affinities. It will be appreciated that aspects can be implemented in any convenient form. For example, aspects may be implemented by appropriate computer programs which may be carried on appropriate carrier media which may be tangible carrier media (e.g. disks) or intangible carrier media (e.g. communications signals). Aspects may also be implemented using suitable apparatus which may take the form of programmable computers running computer programs. Figure 1 is a schematic illustration of a system for generating values for parameters of a model defining a relationship between one or more biological processes of a biological system; Figure 2 is a schematic illustration of a computer suitable for performing processing such as the data processing of Figure 1; Figure 3 is a flowchart showing processing to generate values for parameters of the model of Figure 1; Figure 4 illustrates a ground truth of tumour size over time, a prediction of tumour size based on a quantitative systems pharmacology model fit to three measurements of tumour size, and a prediction of tumour size based on the model of Figure 1 generated using the process of Figure 2. Figure 5 illustrates the mean prediction error between a ground truth of cancer growth rate and a predicted cancer growth rate for a model generated using a conventional modelling approach and the model of Figure 1 generated using the process of Figure 2 as a function of the number of samples used to train the respective model. Figure 6 illustrates the mean prediction error between a ground truth of cancer initial number and a predicted cancer initial number for a model generated using a conventional modelling approach and the model of Figure 1 generated using the process of Figure 2 as a function of the number of samples used to train the respective model. Figure 7 illustrates the mean prediction error between a ground truth of cancer carrying capacity and a predicted cancer carrying capacity for a model generated using a conventional modelling approach and the model of Figure 1 generated using the process of Figure 2 as a function of the number of samples used to train the respective model. Figure 8 illustrates the output values of a generative process governing tumour size reflecting the ground truth of a biological system, and experimental data points measured based on the biological system, based on a specific condition. Figure 9 illustrates predictions of tumour size under the same conditions as in Figure 8 using a base model fit to the experimental data points and the model of Figure 1 generated using the process of Figure 2. Figure 10 illustrates predictions of tumour size under different conditions to those of Figure 8 using a base model fit to the experimental data points of Figure 10 and the model of Figure 1 generated using the process of Figure 2. Referring to Figure 1, experimental data 101 and model data 102 are received and processed by a computer 103 to generate output parameters 104 providing parameter values of the model data 102. As described in further detail below, the computer 103 processes the experimental data 101 and model data 102 using a combined modelling approach in which an initial set of parameter values are determined by fitting the experimental data 101 to the model data 102, and subsequently processing the initial set of parameter values and the model data 102 using a regression model to determine updated parameter values. Such a hybrid approach in which model parameter estimates are first determined in a conventional way by fitting model data to experimental data and then using a regression model to determine updated parameter values for the model has been found to significantly outperform existing techniques for model parameter determination, for example as used in model calibration. The experimental data 101 comprises a plurality of data sets, each data set is defined by an output value and a set of experimental conditions. Each output value provides a value for a property of a biological system under the respective set of experimental conditions for each of one or more sampling intervals defined by an independent variable t, for example for each of a plurality of times t. The independent variable t may be time, or alternatively may be any other sampling interval such as pH, concentration of a particular molecule or the like. The experimental data 101 may be considered as sampled from a generative process denoted ^(^; ^)with ^ =(^^, … , where ^^, ^ = 1 … ^ specifies the set of experimental conditions, and different values of one or more of the ^^, ^ = 1 … ^ defines a different data set of the plurality of data sets. Each data set of the experimental data 101 may be denoted ^ = (^^, … , ^^), where T is the number of sampling values, and is associated with a corresponding set of values of the set of experimental conditions ^. The property of the biological system may be any suitable property such as chemical concentration, density, absolute number of cells or biomolecules that are determined for each set of experimental conditions. The experimental conditions may also be any suitable conditions associated with the biological system. The experimental data 101 may comprise measurements obtained from a real world experiment or may be outputs from a computational model that is based upon real world experimental data. Given the complexity of many biological systems it will be appreciated that obtaining such experimental data, whether by carefully configured real world experiments or from computational models, can be complex. Further, data generated from such computational models can be inaccurate and difficult to interpret given their complexity. The model data 102 represents a relationship between one or more biological processes of the biological system. As noted above, such biological systems are typically highly complex systems with a large number of operating conditions that interact with one another to define the biological system. The model data 102 may, for example, be a minimal mechanistic model that describes the biological system in a simplified form, for example a form that is more easily interpretable. Where the generative process ^(^; ^) used to generate experimental data 101 is a computational model, the model data 102 may, for example, model the biological system with less molecular detail than the generative process. The model data 102 may have the general form ^(^) , … , ^^(^)), with D components ^^(^) ; ^), where ^ =(^^, … , ^^)is the set of N parameters and t denotes the independent variable associated with the experimental data. It can be seen that model component ^^(^)is a function of ^(^)such that ^^(^)may be a function of other model components ^^(^), ^ ≠ ^. While in the below the model data 102 will be described with respect to an independent variable t, it will be appreciated that the techniques described herein are also applicable to model data that does not have an independent variable. As noted above, the experimental data 101 and the model data 102 are both associated with the same biological system. The model data 102 is defined such that a model component of the model data corresponds to the output value of the experimental data 101. For example, where model component ^^(^)of the model data 102 is the model component that corresponds to the output value of the experimental data 101, each data indicates ^^with a noise component, which may be additive noise or multiplicative noise. It should be noted that the noise component and its properties are not required to be known or understood for the modelling described herein. The collective experimental data set =^^^(^^), … ^^(^^)^comprises all single data sets ^^= ,, each with T sampling values, each generated from a distinct set ^^= (^^^, … ^^^), ^ = 1 … ^ where S is the total number of distinct sets of conditions for which data has been generated and M is the number of conditions that are required for each single data set. The number of data points in the collective data set is ^ ∗ ^, i.e. the number of distinct conditions ^ times the number of sampling points ^. As described above, computer 103 processes the experimental data 101 and model data 102 using a combined modelling approach to generate output parameters 104. The combined modelling approach first fits the experimental data 101 to the model data 102 using conventional model fitting approaches to determine a respective set of model parameter values ^^^ … , ^^^^^for each data set ^^such that the model component ^^is a good approximation of the data ^^for the data set ^^. Each respective set of model parameter values ^^^= ^^^^^, … , ^^^^^ define the fit of the model to the corresponding data set ^^. Put another way, the model data 102 provides a good fit to the data points of a given data set ^^when assigned with the model parameter values Each set of model parameters values ^^^may be determined by defining a cost function ^(^^^)such that a set of parameter values is obtained by numerically minimising the cost function, i.e. ^^^= arg min ^(^^). In one example the ^ sum of squares may be minimised such that ^(^^)=∑^^^^^^^^^^; ^^^ − ^^^^ . The fitting of the experimental data 101 provides S sets of parameter data sets comprising each set of model parameter values ^^^=^^^^^, … , ^^^^^, ^ = 1 … ^, (i.e. one set of model parameters for each data set ^^), which is denoted∑^ herein. An individual collective parameter data = ^^^^, … , ^^^^denotes the set of S inferred values of the j-th individual model parameter ^^^for all individual data sets. Each set of model parameter values ^^^=^^^^^, … , ^^^^^provides a parameterisation of the model data 102 that is a good fit to the particular experimental data, however as discussed in further detail below, it has been found that the individual sets of model parameters are generally poor at predicting ground truth data (i.e. additional experimental data that does not form part of the determination of the initial model parameters). As such, according to the methods described herein, the set of model parameter values derived from the experimental data 101,∑^, is further processed using a regression model, for example a Bayesian regression model, that effectively learns the relationship between the experimental conditions ^^and the corresponding sets of model parameter values ^^^, that is, that determines a function ℎ^such that ^^~ℎ^(^^, … , ^^)= ℎ^(^), based on the complete set of parameter values∑^, and corresponding experimental conditions ^^. For each model parameter ^^a regression model may be trained using the collective parameter data set ∑ ^^= ^^^^, … , ^^^^, ^ = 1 … ^ and corresponding sets of experimental conditions ^^. For example, the set of experimental conditions ^^may be used as input and associated model parameter value ^^^^as target output for each ^ = 1 … ^ to obtain a training model ℎ^(^)for all j. An updated value for each model parameter ^^may be obtained based upon the trained model ℎ^(^)for all j to provide the output parameters 104. For example, values for the model parameters may be determined by determining a mean value for each model parameter ^^from the respective trained models. By further processing the model parameters in combination, for example using a regression model, information from each set of experimental data is effectively combined to improve the model parameters such that the model data 102 and associated output parameters 104 generated using the hybrid techniques described herein are able to predict ground truth data more effectively. In some embodiments iterative processing may be performed using outputs of the previous iteration to further improve the output model parameters. For example, output of a regression model as described above may be used as a constraint for further parameter inference. For example, an upper and lower bound may be determined for each model parameter ^^that is used as a constraint on the model, for example as a constraint on the minimisation of a cost function as described above. The processing may be terminated based upon any suitable stopping condition, for example after a predetermined number of iterations, or if the estimated change in model parameters between iterations is below a threshold value. For example, in some embodiments the regression model may be a Gaussian Process model ^^ such that ℎ^(^)= ^^ ^^^(^), ^^^^(^, ^′)^, where ^^(^)is the mean function and ^^^^(^, ^′)is the covariance function for the j-th model parameter ^^. The standard deviation of parameter ^^, ^^(^), is given by the covariance function ^^(^)=^^^^^(^, ^′). Upper and lower bounds for ^^can be defined, for example, as ^^^= ^^(^)+ ^^^(^)and ^^^= ^^(^)− ^^^(^)respectively, where ^ > 0 denotes a hyperparameter of the method, which may be determined, for example, by initialising ^ with a value 1 and increase ^ until the minimization converges. In some embodiments ^ = 2 . Using the constraints ^^^and ^^^, a new estimate for the set of model parameters can be obtained using the experimental data 101 and model data 102, for example by minimising a constrained cost function ^^(^) ^ = ^arg min ^(^^)| ^^^(^^)< ^^< ^^^(^^), ^ = 1 … ^^^^^ ^^^ ^ = 1 … ^. Figure 2 shows a computer 200 suitable for processing the experimental data 101 and model data 102 such as computer 103 of Figure 1. It can be seen that the computer comprises a CPU 200a which is configured to read and execute instructions stored in a volatile memory 200b which takes the form of a random access memory. The volatile memory 200b stores instructions for execution by the CPU 200a and data used by those instructions. The computer 200 further comprises non-volatile storage in the form of a hard disc drive or a solid state drive 200c. The computer 200 further comprises an I / O interface 200d to which is connected peripheral devices used in connection with the computer 200. More particularly, a display 200e is configured so as to display any output from the computer 200. Input devices are also connected to the I / O interface 200d. Such input devices may include a keyboard 200f and a mouse 200g which allow user interaction with the computer 200, or, for example, a touch screen interface. The input devices (200f, 200g) allow a user or an operator to interact with the system 100, for example to indicate training data to be processed. A network interface 200h allows the computer 200 to be connected to an appropriate computer network so as to receive and transmit data from and to other computing devices. The CPU 200a, volatile memory 200b, hard disc drive 200c, I / O interface 200d, and network interface 200h, are connected together by a bus 200i. Figure 3 shows the processing steps of a hybrid modelling approach performed at computer 103 to generate output parameters 104. The processing steps performed by the computer 103 uses the experimental data 101 and model data 102 as previously described to provide output parameters 104 also described herein. At step 300, the computer 103 receives experimental data 101 associated with a biological system. The experimental data 101 comprises a plurality of data sets each defined by an output value ^^=(^^^, … , ^^^)representing T sampling values of a property of the biological system under a respective experimental condition ^^= (^^^, … ^^^), ^ = 1 … ^, where S is the total number of distinct sets of conditions for which data has been generated and M is the number of conditions that are required for each single data set. The experimental data 102 may be obtained based on physical experiments conducted on a biological system or may be generated based on a generative process ^(^; ^) as previously described. At step 302, the computer 103 receives model data 102 which relates to the same biological system as the experimental data. The model data 102 represents a relationship between one or more biological processes of the biological system and may have the general form ^(^) , … , ^^(^)), with D components ^^(^) = ; ^), where ^ =(^^, … , ^^)is the set of N parameters and t denotes the independent variable associated with the experimental data. At step 304, the computer 103 fits the model data 102 to each of the data sets in the experimental data 101. The model data 102 is defined such that a model component of the model data corresponds to the output value of the experimental data 101. The computer 103 uses a conventional model fitting approach to determine a respective set of model parameters ^^^= ^^^^^, … , ^^^^^ for each data set ^^. The conventional model fitting approach may be the cost function minimization technique described previously, where for example the cost is the sum of squared errors between the one of the components of the model data 102 and the output values ^^=(^^^, … , ^^^). Alternatively, any other conventional model fitting approaches may be used to perform parameter inference such a marginal maximum a posterior estimation using Markov chain Monte Carlo or variational inference. At the end of step 304, the computer 103 outputs a parameter set∑^ comprising S sets of parameter data sets for each respective data set ^^and corresponding experimental At step 306, the computer 103 performs processing on the parameter set∑^ in order to obtain a distribution for each parameter ^^of the set of N parameters of the model data 102. The computer 103 uses the regression model as previously described to which may be trained for each model parameter ^^using the collective parameter data set ∑ ^^= ^^^^, … , ^^^^, ^ = 1 … ^ and corresponding sets of experimental conditions ^^. For example, the set of experimental conditions ^^may be used as input and associated model parameter value ^^^^as target output for each ^ = 1 … ^ to obtain a training model ℎ^(^)for all j. In some embodiments the regression model may be a Gaussian Process model ^^ as previously described. At the end of step 306, the computer 103 determines a value for each model parameter ^^based upon its respective trained model ℎ^(^)for all j. For example, as previously described, a value for each model parameter ^^may be determined based on the mean value for each model parameter ^^derived from the respective trained models . The computer 103 may then proceed to step 308 to provide output parameters 104 corresponding to the values derived for each of the respective model parameter values Alternatively, the computer 103 may return to step 304 in order to further improve the values determined for each of the model parameters ^^. The computer 103 uses the trained models ℎ^(^)associated with each of the model parameters ^^in order to constrain the model 102 when fitting of the model data 102 to the experimental data 101 using the conventional model fitting approach. A lower and upper bound ^^^and ^^^is determined for each of the parameters ^^^based on the associated trained model ℎ^(^). For example, a lower and upper bound respectively may be calculated based on the mean of ^^= ^^(^) and the standard deviation of ^^= ^^(^). Upper and lower bounds for ^^can be defined, for example, as ^^^= ^^(^)+ ^^^(^)and ^^^= ^^(^)− ^^^(^)respectively, where ^ > 0 denotes a hyperparameter of the method, The computer 103 outputs the constrained parameter set∑^ and proceeds to step 306 to perform further processing on the constrained parameter set∑^ to obtain updated functions ℎ^by training a regression model on the constrained parameter set∑^ and the set of experimental conditions ^^as previously described. At the end of step 306, the computer 103 determines an updated value for each model parameter ^^based upon its respective updated trained model ℎ^(^)for all j. The computer 103 may then compare the updated values for the model parameters ^^with the values for the model parameters ^^obtained from the previous iteration. If the difference between some or all of the updated values and the previous values is less than a threshold, the computer 103 proceeds to step 308 to provide output parameters 104 corresponding to the updated values for the model parameters ^^. Alternatively, if the different between some or all of the updated values and the previous values is less than a threshold, the computer 103 may then again return to step 304 in order to further improve the values for the model parameters ^^using the most recently updated trained model ℎ^(^)to constrain the model 102. In this way, iterations of steps 304 and 306 can be performed until the values for the model parameter ^^converge to an acceptable amount. Turning to Figure 4, it can be seen that predictions from the model data 102 that has been fit using steps 300-308 described herein more closely reflects the ground truth of a biological system, which in this example is an immuno-oncological system, when compared to a quantitative systems pharmacology (QSP) model which although fitting well to the data points, is a poor predictor of the ground truth of the system, The curve corresponding the QSP model is similar to a curve produced by the model data 102 based on the set of parameter values ^^^=^^^^^, … , ^^^^^corresponding to condition ^^obtained using the conventional modelling approach of step 304. This is due to both the QSP model and the model data 102 at stage 304 being fit based on data points reflecting a specific condition ^^. By contrast, the updated parameter values obtained from stage 306 are less likely to be overfit to any particular condition ^^. The combined modelling approach described herein may be applied to any suitable biological system to determine the parameters of a model defining a relationship between one or more biological processes of the biological system. The biological system may relate to an animal model such as a mouse or may relate to an in vitro model such as an organoid, tumour-on-a-chip or cell culture. The experimental data 101 comprises a plurality of data sets each defining an output value ^^corresponding a property of a biological system under a respective set of experimental conditions. The model data 102 relates to the same biological system as the experimental data 101 and represents a relationship between one or more biological processes of the biological system and defines a relationship between a set of D components ^(^) = (^^(^), … , ^^(^)) and N parameters ^ = (^^, … , ^^). The hybrid modelling approach outputs a set of output parameters 104 which enable the model data 102 to more accurately reflect the ground truth of the biological system when compared to conventional quantitative systems pharmacology approaches. In one embodiment, the combined modelling approach is applied to determine a relationship between biological processes of an immuno-oncological system of an in vivo or in vitro model. Examples for suitable biological processes of an immuno- oncological system that can be modelled using the combined modelling approach are described in Chelliah, Vijayalakshmi, et al. "Quantitative systems pharmacology approaches for immuno-oncology: adding virtual patients to the development paradigm." Clinical Pharmacology & Therapeutics 109.3 (2021): 605-618. As shown in Figure 3 of Chelliah, Vijayalakshmi, et al, within the scientific literature there are a large variety of models capturing the biological processes governing cancer and immune system dynamics. The biological processes captured by any one of the referenced models may be suitably modelled using the combined modelling approach described herein. The biological processes of the immuno-oncological system may relate to one or more of the following processes including immune activation and suppression, tumour vascularisation and tumour cell death. The output values ^^of the experimental data may relate to one of tumour size, immunosuppression, or TME activation. The experimental conditions may relate to the concentrations or amounts of one or more immunotherapeutic drugs or immunotherapies either administered to the animal or applied to the in vitro system. The immunotherapeutic drugs may include one or more monoclonal antibodies, PD-1 / PD-L1 and CTLA-4 checkpoint inhibitors, cytokines, interferons, interleukins or growth factors. The immunotherapies may include oncolytic viruses, CAR T-cells, Bacillus Calmette Guerin (BCG), or a cancer vaccine. The experimental conditions may further relate to the concentration of metabolic factors, oxygen or acidity. The components ^^(^)of the model data 102 may relate to, in either an explicit or approximated way, at least one of tumour size, carrying capacity of the tumour, immunosuppression, TME activation, numbers or concentrations of specific cell types, such as tumour cells, innate and adaptive immune cells, and stromal cells in various compartments of the biological system such as the TME or lymph node, the numbers of specific cell surface receptors on any of the aforementioned cell-types, and the concentration of cytokines secreted by innate or adaptive immune cells. The set of parameters ^ may correspond to at least one of a tumour growth rate, a tumour killing rate of one or more innate or adaptive immune cells, a migration rate of one or more of the various cell types to the TME, and the proliferation rate, activation rate and / or death rate of one or more cell types,. In one embodiment, the combined modelling approach is applied to determine the relationship between expression of one or more genes and / or the downstream effect of such gene expression and one or more biological processes. For example, the combined modelling approach may be applied to determine levels of expressed genes and / or quantities of one or more gene products, such as RNA or proteins, in an in vivo or in vitro model. The biological processes may relate to one or more of the transcription of DNA into RNA, translation of RNA into proteins, and degradation of RNA and / or protein molecules. The output values ^^of the experimental data may relate to copy numbers or concentrations of one or more RNA transcriptions or the numbers or concentrations of one or more proteins. The experimental conditions may relate to concentrations of one or more pharmaceutical drugs either administered to the animal or applied to the in vitro system. The components ^^(^) of the model data 102 may relate to, in either an explicit or approximated way, RNA transcript copy number or concentration corresponding to one or more genes and numbers or concentrations of one or more proteins, transcription factors and cofactors. The may correspond to at least one of binding affinities between specific transcription factors and an associated binding site, a transcription rate for one or more genes, a facilitatory effect on the transcription of one or more genes by one or more cofactors, an inhibitory effect of on the transcription of one or more genes repressors, and activatory or inhibitory effects relating to chromatin structure and orientation of binding sites. In one embodiment, the combined modelling approach is applied to determine the relationship between the bioproduction of one or more biomolecules and one or more biological processes in cell lines. The output values ^^of the experimental data may relate to the number or concentration of a biomolecule such as a protein, viral vector or virus like particles (VLP). The protein may be a therapeutic protein. The experimental conditions may be one or more of concentrations of chemical inducers, transfection reagents, nutrients, plasmids, RNA transcripts. The components ^^(^)of the model data 102 may relate to, in either an explicit or approximated way, the number or concentration of the produced biomolecule and the number or concentration of the cells used for expression. The parameters ^ may correspond to at least one of cell growth rate, cell death rate, a rate of transcription and / or translation, and transfection efficiency. Application to cancer growth The implementation of the combined modelling approach will now be described in relation to cancer growth. As described above, it will be appreciated that the combined modelling approach is applicable to various biological systems and the below is provided as an illustrative example of the techniques only. Mechanistic Model We use a nested logistic growth model as a model data 102 for cancer growth, defined by the following ODE initial value problem: ^ ^^(^) ^^(^) = ^^^^(^)(1 − ); ^ ^^(0) = ^^^(^) 3 2 Here ^^denotes the number of cancer cells, and ^^a (time-dependent) carrying capacity with fixed growth rate. We make this choice to account for immunosuppressed cancer growth at onset (i.e. the effective exponential growth rate increases). The mechanistic parameters of the model are the (static) cancer growth rate ^^, the initial number of cancer cells ^^, and the maximal (final) carrying capacity ^^. We assume that only the number of cancer cells is [experimentally] observed. For the ODE system specified above we can calculate the analytical solution: Generative Process Our aim is to construct a generative process that emulates either an experiment or a different [mechanistic] model, given the underlying conditions ^ = (^^, ^^, ^^), and their relationship to the mechanistic parameters ^ = (^^, ^^, ^^). To that end, we fist define a “response function” 1 1 ^^(^, ^, ^, ^) = ^(tanh(^(^ − ^)) + 1) + 2 2 And then the following functions that [heuristically] encode the effect of certain chemical compounds ^^, ^^, ^^on the cancer-immune interactions, that are either tested in an experiment or expressed in another model (i.e. either case being represented by the generative process below). ^isAct(^^) = ^^(^^, 0.5,2,20) ^isSupp(^^) = ^^(^^, 0.3,4,2) ^tmeAct(^^, ^^) = ^^(^^, 0.1,4,2)^^(^^, 0.3,5,3) The particular choices for ^, ^, ^ ensure we obtain nonlinear responses defined by a mixture of intrinsic scales ∼ ^, ^ and transition points ∼ ^. The underlying rationale is such that ^isAct(^^) mimics the effect of immune system activation and depends on ^^, ^isSupp(^^) mimics the effect of immune system suppression regulated by ^^, and ^tmeAct(^^, ^^) mimics the growth stimulation via formation of the tumour microenvironment (TME) and depends on ^^and ^^. These processes define the predominant mechanistic effects of the chemical compounds ^^, ^^, ^^. With these definitions at hand we define the relationship between mechanistic parameters and [experimental] conditions as follows: ^ ) where ^^ ^ set the respective reference scale for that parameter. Using the analytical expression for ^^(^) we construct the generative process Virtual Experimental Study We now use the generative process to generate where ^ ∼ ^[0, ^] is a normally distributed random variable with zero mean and ^ standard deviation. Unless noted otherwise, we will use ^ = 0.15 to obtain data samples with a coefficient of variation of 0.15. Parameters For the virtual experimental study we sample from the generative process, where the values of each ^^are taken from the set {0.1, 0.3, 0.5, 0.7, 0.9, 1.0, 3.0, 5.0, 7.0 , 9.0}, i.e. corresponding to ^ = 10^= 1000 combinations. We note that the method achieves similar performance as reported below with sparse sampling of the same range of values, e.g. with ^ = 5^= 125 (every second value from above range dropped). We consider the interval ^ = [0 … 12] equidistantly partitioned to obtain ^ sampling points ^^, where we will use different values of ^ to investigate how the method performance depends on data size. Furthermore we set the following reference scales for the parameters 0.3, ^^ ^ = 1.0, ^^ ^ = 5 Hybrid AI Loop With above specification of the mechanistic model and the generative process, we run steps 300-308 of the combined modelling as previously described and perform n = 10 iterations of steps 304 and 306. Here the initial iteration fits the model ^^(^) to the experimental data 101 comprising output values ^^that are specified by ^^for all S = 1000 possible combinations of (^^, ^^, ^^). For the technical implementation we use Wolfram Mathematica 13.2. We use the build- in function FindFit[] with “NMinimize” as minimization method to minimize the ^^Norm (sum of squares) between mechanistic model and data (as described for step 304) in each iteration step. For step 306, the AI component uses the build-in Predict[] function, with “GaussianProcess” as fixed method and “SquaredExponential” as covariance function. As iterative constraint we use the AI components as described in the technical documentations with ^ = 2 as scaling factor. Note that at no point did we use any information about the functions ℎ^defined above. It is only used to generate the data and the aim is to infer exactly these relationships from the data. Results and Method Performance Metrics Evaluation of Method Performance To evaluate the performance of the method, we compare the mechanistic parameters obtained from the initial fit (no constraint from the AI components) to the ~ parameters obtained after 10 iterations of the hybrid AI method ^(^^) ^^ , ^ ^ = 1,2,3, in each case we calculate the error (residuals) relative to the actual true parameters for that case defined by ℎ^(^^). For each parameter ^ = 1,2,3 we calculate the mean of the (normalized) errors for the ^-th iteration: ^ where ℎ^(^^)is the ground truth parameter for each combination ^^as defined above. This highlights why we needed to define the generative process in this virtual experiment the way we did: it enables us to unambiguously define the effect of the hybrid AI iteration on parameter inference by comparing the true value with the inferred value. Of course, in any typical real world experimental scenario we would not have access to the true value, and our ability to assess the performance of the method would be significantly limited. Hence, the purpose of the validation here is to assess the quality of results that could be expected in any application of the method to real data. General performance metrics First we look at the mean prediction error aggregated over the entire data set for all ^ as a function of data size ^, which is shown in Figures 5, 6 and 7. As shown in Figure 5, for the cancer growth rate ^^we observe a 100-fold reduction in the relative prediction error at low sampling densities ^ = 3, obtaining a sensitivity of ∼ 15%. Achieving similar results without the hybrid AI method requires increasing the data size ^ more than 20-fold. However, even at high sampling densities, the hybrid AI methods outperforms the conventional method by a factor 5x to 10x. As shown in Figure 6 and 7, for both cancer initial number ^^and cancer carrying capacity ^^we observe that the improvement remains constant for increasing ^, indicating that the hybrid AI method constitutes a qualitative improvement that cannot be achieved by conventional methods, even if increasing data size. Individual Examples To better understand the underlying mechanism of the hybrid AI method, it is instructive to look at individual examples. We start with the generative process shown in Figure 8. The dashed curve shows the generative process for the given values of ^ =(0.7,7.0,3.0). The data is sampled from this process and perturbed by multiplicative noise ^. Let us then compare model fits without AI (base model), and the hybrid ~ AI inference ^(^^) ^ (combined modelling) with the respective ground truth. As shown in Figure 9, we can immediately observe that the original fit without hybrid AI (base model) has minimal residuals. It is an optimal fit of the data, and conventional methods wouldn’t be able to improve it based on the three available data points alone. The hybrid AI (combined modelling) method on the other hand expresses correlation between nearby conditions ^, ^^via the covariance function ^(^, ^^), (see the Gaussian Process model ^^ described earlier). This prevents overfitting in sparse data regimes as here. Let us compare this result with a nearby condition. As shown in Figure 10, again, we can see that the base model shows minimal residuals, but overfits the data. Compared to the previous case, the change in the underlying ground truth model is much smaller than the change suggested by the sampled data points, and again this small change is captured by the hybrid AI model (obtained using the combined modelling approaches described herein) with high accuracy.

Claims

CLAIMS:

1. A computer-implemented method of generating values for parameters of a model defining a relationship between one or more biological processes of a biological system, the method comprising: receiving a plurality of data sets, each data set associated with a property of the biological system under a respective set of experimental conditions of a plurality of sets of experimental conditions, each data set associating a respective output value for the property of the biological system with the respective experimental conditions; receiving a model defining a relationship between the one or more biological processes of the biological system, the model comprising a plurality of model components, each model component of the plurality of model components defining a respective relationship between a plurality of model parameters, wherein a model component of the plurality of model components defines a relationship between the plurality of model parameters and the property of the biological system; for each model parameter of the plurality of model parameters, determining a plurality of initial model parameter values, each value of the plurality of initial model parameter values associated with a respective set of experimental conditions and determined based on a respective data set of the plurality of data sets; and for each model parameter of the plurality of model parameters, processing the plurality of initial model parameter values and the respective set of experimental conditions to generate updated model parameter values.

2. A method according to claim 1, wherein processing the plurality of model parameter values and the respective set of experimental conditions to generate updated values for the plurality of model parameters comprises: training a regression model based on the plurality of model parameter values and the respective set of experimental conditions.

3. A method according to claim 1 or 2, wherein processing the plurality of model parameter values and the respective set of experimental conditions to generate updated values for the plurality of model parameters comprises: generating one or more bounds for the values of the model parameter; and determining an updated plurality of model parameter values based on the respective data set of the plurality of data sets and the one or more bounds.

4. A method according to claim 3, wherein the one or more bounds for the values of the model parameter comprise a range of values for the model parameter.

5. A method according to claim 4, wherein the range of values for the model parameter comprises an average value and a standard deviation.

6. A method according to any one of claims 3 to 5, determining an updated plurality of model parameter values based on the respective data set of the plurality of data sets and the one or more bounds comprises: for each model parameter of the plurality of model parameters, determining a plurality of updated model parameter values, each value of the plurality of updated model parameter values associated with a respective set of experimental conditions and determined based on a respective data set of the plurality of data sets.

7. A method according to any preceding claim, further comprising fitting the model to a respective data set of the plurality of data sets to determine a respective initial model parameter value of the plurality of initial model parameter values for each model parameter of the plurality of model parameters.

8. A method according to claim 7, wherein fitting the model to a respective data set of the plurality of data sets comprises minimising a cost function associated with the model component of the plurality of model components that defines a relationship between the plurality of model parameters and the property of the biological system.

9. A method according to any preceding claim, wherein the plurality of data sets is obtained based upon experimental data.

10. A method according to any preceding claim, wherein the number of model parameters is smaller than the number of experimental conditions.

11. A method according to any preceding claim, wherein the model defines a relationship between expression of one or more genes and one or more biological processes of a biological system.

12. A method according to any preceding claim, wherein the model defines a relationship between biological processes of an immuno-oncological system.

13. A method according to any preceding claim, wherein the model defines a relationship between the bioproduction of biomolecules and one or more biological processes in a cell line.

14. A carrier medium carrying computer readable program code configured to cause a computer to carry out a method according to any one of claims 1 to 13.

15. A computer apparatus for generating values for parameters of a model defining a relationship between one or more biological processes of a biological system, the apparatus comprising: a memory storing processor readable instructions; and a processor configured to read and execute instructions stored in the memory; wherein said processor readable instructions comprise instructions controlling the processor to carry out a method according to any one of claims 1 to 13.

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