Sensitivity analysis of dynamic biological systems with time-delays.
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These systems are represented as delay differential equation (DDE) models.The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors.To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary.
Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw
ABSTRACT
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Background: Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary. Results: We have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention. Conclusions: By comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays. |
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Mentions: The EAMCM program is used to do sensitivity analysis on the lumped cardiovascularmodel. The non-constant exponent of Hill function and sinusoidal functions indifferential model equations complicate the evaluation of Jacobian matrix forcomputing the solution of sensitivity equations. By the help of automaticdifferentiation embedded in the EAMCM program, user can provide the model equationsonly at run-time for solving the dynamic sensitivities of the cardiovascular system.The dynamic sensitivities of heart rate and blood pressure with respect to all systemparameters and initial conditions are computed. All relative parameter sensitivitiesare presented by 100% stacked column chart and shown in Figure 1. It is easy to find which parameter makes more effects on heart rate andblood pressure than the others from Figure 1. The values of topfive sensitivities for the heart rate and blood pressure are shown in Table 2. The uncontrolled average arterial blood pressure(p0), breathing rate (fr),sympathetic delay (τ), sympathetic control of heart rate(β), and strength of vagal tone (ν) are identified asthe key parameters for the control of heart rate and blood pressure. The relativesensitivities of heart rate and blood pressure with respect to the uncontrolledaverage arterial blood pressure are shown in Figure 2. Thedynamic sensitivities of heart rate with respect to p0 oscillatesymmetrically between positive and negative values. This result indicates that theuncontrolled average arterial blood pressure amplifies the variation of heart rate.In contrast, the dynamic sensitivities of blood pressure with respect top0 oscillate but are all positive. This means that an increaseof the uncontrolled average arterial blood pressure shifts the blood pressure to ahigher value but does not change the variation of blood pressure. As shown in Table2, the effect of uncontrolled average arterial bloodpressure on the variation of average heart rate is tenfold larger than the variationof blood pressure. There is evidence that the slow-acting sympathetic nerves and thefast-acting vagal nerves compete with each other to increase and decrease the heartrate, respectively [36]. The relativesensitivities of heart rate with respect to parameters for slow sympathetic control(β) and fast vagal control (ν) are investigated andshown in Figure 3. Figure 3 shows thesympathovagal balance in physiology and both sympathetic control and vagal controlamplify the variation of heart rate. The relative sensitivities of blood pressurewith respect to parameters for slow sympathetic control (β) and fastvagal control (ν) are shown in Figure 4. Theslow-acting sympathetic control upregulates the blood pressure, but does not changeits variation. The relative sensitivity of blood pressure with respect to thesympathetic control is positive over the time. In contrast, the fast-acting vagalcontrol downregulates the blood pressure and has a negative relative sensitivity overthe time. |
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Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw
Background: Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary.
Results: We have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention.
Conclusions: By comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays.