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Parametric Resonance Applications in Neutrophil Dynamics | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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Parametric Resonance Applications in Neutrophil Dynamics

Suqi Ma1 and Jinzhi Lei2*

1Department of Mathematics, Chinese Agricultural University, Beijing 100083, China

2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100083, China

*Corresponding Author:
Jinzhi Lei
Zhou Pei-Yuan Center for Applied Mathematics
Tsinghua University, China
Tel: 86 10 6279 3001
E-mail: [email protected]

Received Date: December 08, 2015; Accepted Date: January 19, 2016; Published Date: January 25, 2016

Citation: Ma S, Lei J (2016) Parametric Resonance Applications in Neutrophil Dynamics. J Phys Math 7:154. doi:10.4172/2090-0902.1000154

Copyright: © 2016 Ma S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The profound effects of chemotherapy on the combined dynamics of the hematological stem cells and their differentiated neutrophils is examined. G-CSF is often used to deal with this neutropenia and the response is highly variable. To shape the neutrophil response to chemotherapy and G-CSF, periodic parametric resonance is discussed. Periodic oscillation in neutrophil levels and the subharmonic 1:2 resonance phenomena are observed with the assumption of periodic chemotherapy is given. The work is aim to stimulate further investigations and the practical applications


Parametric resonance; Neutrophils; Hematological stem cells


In the treatment of malignant tumors, chemotherapy is frequently accompanied by hematopoietic side effects due to the myelosuppressive character of drugs used. A common neutropenia(accompanied by fever and possible infection), to a lesser extent, thrombocytopenia and anemia are frequently observed. To circumvent these side effects, an effective administration of granulocyte colony stimulating factor (G-CSF) is suggested in clinic practice [1-7], and now a standard postchemotherapy treatment for neutropenia. Mathematically, people try to understand G-CSF control by a variety of models [3,8-12].

A significant G-CSF support in clinic perspective is periodically repeated treatment of chemotherapy, and neutrophil is highly responsive and dependent on timing and protocol of the drug’s administration [13,14]. Experiment have reported that chemotherapy increases apoptosis in both proliferative HSCs (hematopoietic stem cells) and proliferative neutrophil precursors [15]. To idealize the effects of chemotherapy and G-CSF, a square wave temporal functions for the loss rates,neutrophil precursor proliferative rate are used, explained in reference [16]. The resonance between the perturbation due to the periodic chemotherapy and the intrinsic characteristic frequency in the neutrophil production have been discussed. The periodic damped oscillation in neutrophil levels [17] are also reported with the assumption of single dose of chemotherapy, which is simulated by exponential mathematically.

In this paper, we explain the hematopoietic dynamics with the pharmacokinetics of chemotherapy and G-CSF by periodic parametric excitation. The dependence of the neutrophil response on the stimulation frequency is examined and resonance phenomena is discussed.

The Model Equations

The combined dynamics of hematopoietic stem cells and neutrophil production is illustrated in Figure 1 and generally described by the following equations



Figure 1: A cartoon representation of system.

where Holling function


and rates


The neutrophil system (1) consists of DDEs(delay differential equations) with two delays. These two equations takes into account the balance between the net production and loss terms of HSCs and circulating neutrophils, where Q and N (both with unit cells/kg)are respectively hematopoietic stem cells and neutrophils. HSCs differentiate into the committed neutrophil compartment with term kN (N)Q. Delay τs denotes their duration days of proliferative phase of HSCs. HSCs either remain in the resting phase or enter into the proliferative phase at a rate β(Q). The differentiation of HSCs to neutrophil is controlled by kN (N) (the differentiated rate of circulating neutrophils), and HSCs proliferation is controlled by the resting HSCs population with proliferation rate β(Q). rs represents the apoptosis rate during the proliferation phase of HSCs. The circulating neutrophils survive with their averaged lifespan equation and thus with the ature death γN N. Cells in the neutrophil lines experience a successive division with time period τNP and then enter into a purely maturation compartment with time period τNM, and the total time period τNNPNMNP denotes the proliferative rate of neutrophils and γo denotes their death rate before neutrophils enter into the circulation. The parameter values in System (1).

Suppose chemotherapy is administrated periodically. Neutropenia is a common side-effect, and G-SCF is also administrated after chemotherapy. Both the effects of chemotherapy and G-SCF are maintained for several days. Chemotherapy increases apoptosis in both proliferative HSCs and proliferative neutrophil precursors. The perturbation of parameter values are supposed to be


Normally, the rates rsNP and γo are constants, and therefore



The response of neutrophil to chemotherapy, and the effect of G-CSF adminstration always can be strong as varying excitation frequency near natural frequency. Under the excited vibration of parameters rsNP and γo, resonance oscillation can be initiated in the interval of the excitation frequency due to the nonlinearity. It is well known, system can be unstable due to the action of parametric excitation. In certain intervals of frequency Ω lying,system(1) may have phase lock solutions. Set ω1= ω2= ω3=Ω, and then let equation in general, there exist intervals of excitation frequency O with bigger width to induce 1:1 primary resonance or 1:2 subharmonic resonance phenomena. The combination parametric resonance can be triggered which induce the periodic solutions with large amplitude. To perform bifurcation analysis, we choose O as bifurcation parameter, then define equation, and set .equation to be solution flow of system (1) with initial values Q(t-t)= Q*,N(t-t)=N* for equation where τ=maxs, τN). Since Ω:ω=1:1, every point on its orbit is a period point. Define the amplitude || N|| to describe the difference between the maximal period point and the minimal period point. We varied Ω widely and, for each value, solved DDEs (1) for t=1200 days using initial values Q= Q*,N=N* for equation due to delays τsN satisfied. The long term effect of chemotherapy and G-CSF for each simulation the last 400 days are used to obtain the amplitude in neutrophil fluctuations. || N|| continue to a bigger value to form the resonance in a widely varying scope of excitation frequency. The 1:1 resonance regime in parameter space (Ω,τs)plane is formed in a shaded regime as shown in Figure 2a. In special, choose τs =3.2, the response of neutrophil numbers near 1 (the ratio of Ω/ω) are denoted by its amplitude exhibits large values continuously. The amplitude response corresponding to the ratio of Ω/ω is drawn in Figure 2b.


Figure 2: Primary resonance phenomena in system (a) Resonance regime of 1:1 primary resonance formed in parameter space. The shaded regime is the called 1:1 resonance regime. (b)The amplitude response of neutrophils versus varying ratio Ω/ω.

Vary excitation frequency Ω near ω/2 with ω being natural frequency, the 1:2 subharmonic resonance regime in parameter space (,τ) plane is also formed by shaded regime in Figure 3a. In special, chooseτs =3.2, the response of neutrophil numbers near 0.5 (the ratio of Ω/ω) are drawn which exhibits 1:2 subharmonic resonance phenomena as shown in Figure 3b. The amplitude response exhibits large values continuously.


Figure 3: Ubharmonic resonance phenomena in system (a) resonance regime of 1:2 subharmonic resonance formed in (Ω,τ) parameter space. The shaded regime is the called 1:2 resonance regime. (b)The amplitude response of neutrophils versus varying ratio Ω/ω.


In a physiologically realistic point view, a two compartment model combined the dynamics of hematopoietic stem cells and the differentiated neutrophils with its response dependence on the period chemotherapy and G-CSF adminstration are given. The combined parametric excitation resonance phenomena are discussed. When the ratio between excitation frequency and eigenfrequency are rational, both 1:1 primary resonance and 1:2 subharmonic resonance are tracked which respectively form a resonance regime on plane of parameters.


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