Manuel Tetschke (1) , Patrick Lilienthal (1), Torben Pottgiesser (2), Thomas Fischer (3), Enrico Schalk (3), Sebastian Sager (1)
(1) Institute for Mathematical Optimization, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany, (2) Department of Cardiology and Angiology I, Heart Center Freiburg University, Faculty of Medicine, University of Freiburg, 79106 Freiburg, Germany, (3) Department of Hematology and Oncology, Medical Center, Otto-von-Guericke-University Magdeburg, 39120 Magdeburg, Germany
Introduction:
The production of red blood cells (RBCs) in humans is an individual complex process. Our main interest lies in the regeneration of RBCs after a blood loss. The deeper understanding of this regeneration process could have an important impact for personalized clinical decision support in the case of polycythemia vera (PV). PV is a slow-growing type of blood cancer, where especially the production of RBCs is increased. The principal treatment targeting the symptoms of PV is bloodletting (phlebotomy), at regular intervals that are based on personal experiences of the physicians. Due to the complexity of the process, reduction to the most essential features concerning the application is crucial for the development of a model usable in clinical practice.
Objectives:
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Development of a novel simple compartment model for RBC regeneration usable in clinical practice
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Model verification and estimation of characteristic variables for personalization of the model using clinical data
Methods:
In [1] a three compartment model was developed covering the most essential aspects of RBC regeneration after a blood loss in healthy human adults. The essential biological feedback mechanism by erythropoetin (EPO) was included indirectly by a negative feedback expression. Individual aspects of the underlying dynamics are covered by two amplification variables: BETA for the overall cell maturation and GAMMA for the reaction to a blood loss.
The experimental data used in this study were obtained in 2008 in [2]. Here the recovery time of total hemoglobin mass (tHb) after a blood donation in healthy adult was investigated. Therefore, tHb before and after 1-unit (erythrocyte concentrate) standard blood donation was evaluated in 29 male, healthy volunteers (30 +- 10 years, 181 +-7 cm, 76.6 +- 11.2 kg). The use of tHb data ensures a much higher precision than hematocrit (Hct) measurements routinely used in clinical practice. Numerical evaluation was performed on 24 data sets, as five sets were excluded due to unreasonable outliers in the data or a bad initial guess of an initial value.
Both nonlinear mixed-effects estimation and point estimation methods were applied to investigate the model dynamics. First, a nonlinear mixed-effects estimation for the two variables was performed using NONMEM (version 7.4, first-order conditional estimation method with interaction) in combination with PsN software (version 4.4.0; Uppsala Pharmacometrics, Uppsala, Sweden). We used an exponential model for inter-individual variability (diagonal OMEGA matrix) and an additive model for residual variability. Secondly, for point estimation, using the available data and the derived model, parameter estimation problems with a least-squares objective were solved with a multiple shooting based Gauß-Newton algorithm coded in the PAREMERA software and an adaptive, error-controlled backward differentiation formulae (BDF) method for integration coded in the software DAESOL, both included in the experimental design package VPLAN [3] developed at the University of Heidelberg.
Results:
Using the nonlinear mixed-effects estimation, the fixed effects of the variable BETA was estimated as 1.02 +- 0.151, the inter-individual variance as 0.294 +- 0.125. For GAMMA, the fixed effect was estimated as 0.46 +- 0.0651 with an inter-individual variance of 0.346 +- 0.148. The high inter-individual variances of more than 40% for the even quite homogeneous population (male, healthy, non-smokers) suggest the use of point estimation methods. Point estimation lead to a very good fit based on average values for R² of 0.86 +- 0.11. Average values for BETA and GAMMA were 1.519 +- 0.751 and 0.555 +- 0.215, respectively.
Conclusions:
A three compartment model with a negative feedback for erythropoiesis was developed. Essential physiological properties were captured in the model, which could be shown with the application of the RBC regeneration after a blood donation. In this early phase, point estimation methods might be more suitable than population estimation approaches due to heterogenous and sparse data from pathological cases. Point estimation without regularization was successful in most of the cases and can be improved with additional initial information about the subject. Next steps are the evaluation of the model on data for multiple blood donation cycles and extention of the model to PV patients.
References:
[1] Tetschke, M.; Lilienthal, P.; Pottgiesser, T.; Fischer, T.; Schalk, E.; Sager, S. (2018) “Mathematical Modeling of RBC Count Dynamics after Blood Loss.” Processes, 6, 157.
[2] Pottgiesser, T.; Specker, W.; Umhau, M.; Dickhuth, H.H.; Roecker, K.; Schumacher, Y.O. “Recovery of hemoglobin mass after blood donation.” Transfusion 2008, 48, 1390–1397.
[3] Bock, H.G.; Körkel, S.; Schlöder, J.P.: “Parameter Estimation and Optimum Experimental Design for Differential Equation Models.” In Model Based Parameter Estimation: Theory and Applications; Bock, H.G., Carraro, T., Jäger, W., Körkel, S., Rannacher, R., Schlöder, J.P., Eds.; Springer: Heidelberg, Germany, 2013; pp. 1–30.
Reference: PAGE 28 (2019) Abstr 8847 [www.page-meeting.org/?abstract=8847]
Poster: Drug/Disease Modelling - Other Topics