Using a smoother approach IMPRES-M to estimate parameters of biomarkers in multi-layered turnover networks
Jie Ju1, Jeroen Elassaiss-Schaap1, Lorenzo Cifelli1
1PD-value B.V.
Introduction Turnover systems can carry the dynamic response of biomarkers to a drug. Specifically, the production and elimination of the some molecules are directly stimulated or inhibited by a drug, whereas these molecules might subsequently influence the activities of downstream molecules, resulting in a layered network of biomarker turnover. To characterize the network efficiently, we propose to apply a smoother approach IMPRES-M [1], which is used to represent the concentration profile and enables direct estimation of the parameters controlling the production and elimination of the biomarkers in a turnover system. In the current work, we set up simulation methodology to construct layered networks of varying complexity, implement IMPRES-M for efficient parameter estimation and evaluate its accuracy as a function of increasing complexity. Methods For now, we consider a network of turnover models such that the production of a component representing a biomarker is stimulated or inhibited by one upstream molecule (i.e. the drug or another biomarker), defined as dE/dt = kinf(C)-koutE. The E and C are the biomarker and the upstream molecule concentration over time t. kin and kout are parameters controlling molecule productions and eliminations. An algebraic integration of the turnover model was constructed in which the function f represents effect of an upstream molecule, defined as f(t) = 1 – aC(t). The smoother approach IMPRES-M was described in our previous work [1]. Briefly, the function C(t) is represented as a linear combination of B-spline functions. Therefore, we applied the rectangle rule integration and approximated E(t) by evaluating the unknown parameters kin, kout , and a with the R function nlminb [2]. To assess the performance of IMPRES-M, we simulated molecular networks of the turnover systems with four biomarker layers. The first layer of the networks contains biomarkers directly affected by the drug concentration profiles in plasma. Then, we created three additionally layers of downstream biomarkers, and each of them was affected by one corresponding biomarker in the previous layer. All of the biomarker layers include 5 inhibited and 5 stimulated biomarkers, respectively. The accuracy of parameter estimation was evaluated as mean fold change (MFC, in percentage) between observed and predicted parameters. Results The results showed that IMPRES-M effectively estimated the biomarker parameters in the four-layer turnover network. The approach achieved MFC among 40 biomarkers with kin of 0.56% (layer 1-4: 2.1%, 0.12%, 0.011%, and 0.016%), kout of 0.60% (2.2%, 0.13%, 0.0096%, and 0.016%) and a of 57% (66%, 58%, 51% and 55%). Furthermore, we tested the performance of the approach with noise in data. With a standard error of 0.01, the approach yielded increased MFC values in the kin of 8.1% (3.8%, 9.2%, 10%, and 9.3%), kout of 9% (4.1%, 11%, 11%, and 10%) and a of 63% (66%, 59%, 63% and 62%). With a standard error of 0.02, the values of MFC further raised, with kin of 23% (15%, 22%, 22%, and 26%), kout of 18% (10%, 11%, 16%, and 29%) and a of 66% (59%, 71%, 68% and 65%). With a standard error of 0.05, the values of MFC reached the highest, with kin of 40% (13%, 76%, 28%, and 50%), kout of 47% (13%, 76%, 52%, and 54%) and a of 87% (74%, 148%, 70% and 68%). The run time was a under 1 minute for this system of 40 markers. Taken together, we demonstrated that increased residual error result in higher MFC values of estimated parameters, whilst the depth of a layer does not appear to have an impact on the accuracy of the estimations. Conclusion We found that the IMPRES-M accurately estimated parameters for turnover models in layered networks of turnover markers. Moreover, the utilization of this approach improves the computational efficiency in modeling the turnover systems by alleviating the necessity of defining and parameter estimation of ordinary differential equation (ODE) systems. In the future, a more complex relation function of molecule effects f could be included. Furthermore, the approach could be extended to manage the effects on the removal of molecules.
[1] Elassaiss-Schaap, J., Cifelli, L., & Eilers P.H., PAGE 2024, Abstract title: “Construction of IMPRES-M, a non-parametric impulse-response modeling method, in the context of varying pharmacokinetic profiles”. [2] David M. Gay (1990), Usage summary for selected optimization routines. Computing Science Technical Report 153, AT&T Bell Laboratories, Murray Hill.