Physiological and pathological population dynamics of circulating human red blood cells

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Entry by Zin Lin, 09/16/2012 (AP 225 Fall 2012)

General Information

Authors: John M. Higgins, L. Mahadevan

Publication: Higgins et al. Physiological and pathological population dynamics of circulating human red blood cells. Proc Natl Acad Sci U S A. 2010 Nov 23;107(47):20587-92. Epub 2010 Nov 8.

Keywords: Brownian Motion, Langevin Equation, Fokker Planck Equation, Stochastic Differential Equation

Summary

From the time of its birth in bone marrow, an average red blood cell undergoes complex variations in its volume <math>v</math> and hemoglobin(Hb) content <math>h</math>. Immediately after its birth, an RBC undergoes rapid reductions in both <math>v</math> and <math>h</math> followed by a much slower reduction phase. The correlation between <math>h</math> and <math>v</math> improves over time, particularly during the slow reduction phase when the RBC matures towards a well-defined mean corpuscular Hb concentration (MCHC). Based on these empirical observations, Higgins et al. developed a mathematical model which accurately captures the population statistics of human red blood cells. A key assumption of this model is that individual RBC volume and Hb content (<math>h, v</math>) essentially undergoes Brownian motion constrained by an interplay between random fluctuating forces and deterministic dissipative forces. Specifically, the model calculates the joint probability distribution of <math>h</math> and <math>v</math> characterized by a set of dynamical parameters. It is shown that a comparison of these parameters enables one to distinguish between healthy individuals and anemic patients and, more importantly, even identify pre-anemic patients weeks before anemia becomes clinically detectable.

Mathematical Model

The central assumption of the model is that the dynamical variables <math>v</math> and <math>h</math> (RBC volume and Hb content) follows a Brownian-like behavior so that they can be well treated by a Langevin equation with a deterministic drift <math>\boldsymbol{f}</math>and a stochastic drive <math>\boldsymbol{\zeta}</math>. <math>\boldsymbol{f}</math> is further decomposed into fast <math>\beta</math> and slow <math>\alpha</math> components whereas <math>\boldsymbol{\zeta}</math> is modeled as a Gaussian variable with mean zero and a variance given by the diffusion tensor <math>2\boldsymbol{D}</math>:

Image-pic1.jpg

Introducing the RBC birth <math>b(v,h,t)</math> and clearance <math>d(v,h,t)</math> functions, one can write down the Fokker-Planck equation for the time-dependent probability density function <math>P(v,h,t)</math>:

Pic2.jpg.

Here, the clearance function <math>d</math> is modeled on empirical data and is characterized by a clearance threshold <math>v_c</math>, beyond which most RBCs have been cleared; on the other hand, the birth function can be deduced from the condition that at equilibrium, equal numbers of RBCs are being produced and destroyed. One can numerically solve the FP equation but more interesting is the steady state behavior <math>\lim_{t \rightarrow \infty} P(v,h,t) </math> which can be compared with the actual measurements.

Clinical Applications and Discussion

Figure 1: Boxplots of model parameters <math>\beta_v, \beta_h, \alpha, D_v, D_h, v_c</math> for healthy versus anemic profiles. Box edges indicate 75th (upper) and 25th (lower) percentiles whereas the horizontal red lines denote median values. Figure taken from [1]

Notice that the probability distribution is characterized by the set (<math>\boldsymbol{ \alpha, \beta, D, } v_c</math>). With an appropriate choice of these parameters, the authors found that the model faithfully reproduces the observed probability distribution of RBCs in healthy individuals [1]. More importantly, the model can distinguish between healthy and anemic individuals. Figure 1 shows clear differences between the model parameters for healthy individuals and three types of anemic patients: anemia of chronic disease (ACD), thalassemia trait (TT) and iron deficiency anemia (IDA). For example, one can observe that TT and IDA patients lose more of RBC volume and Hb content in slow phase than in rapid phase compared to healthy and ACD individuals (<math>\beta</math> of TT and IDA is lower than that of healthy and ACD whereas <math>\alpha</math> of TT and IDA is higher than that of healthy and ACD). Also, variance of Hb content <math>D_h</math> is significantly higher in IDA patients. Furthermore, the clearance threshold <math>v_c</math> is appreciably lower in IDA, implying that RBC clearance is somehow being delayed in this type of anemia. Remarkably, the authors demonstrated that the latter fact can be used to identify latent or compensated IDA which is otherwise masked by normal Complete Blood Counts (CBC) [1].

As noted in the paper, the actual mechanisms by which RBCs are produced and destroyed or their volume and Hb contents modified are almost impossible to track and are still not well understood. Nevertheless, the broad assumptions of Brownian motion, specifically the concepts of fluctuation and dissipation, turn out to be quite fruitful; not only do they capture the observed population statistics of RBCs but their clinical implications are quite profound. It is remarkable that even though not knowing the details of specific cellular mechanisms, one can get so far based on principles apparently borrowed from a disparate area, the kinetic theory of particle motion. In fact, the picture of <math>h</math> and <math>v</math> behaving like Brownian particles might not be very obvious nor convincing but the insight that biological processes being inherently stochastic, the dynamics of <math>h</math> and <math>v</math> must still be constrained by some fluctuating and dissipative forces (regardless of the biological mechanisms that produced them) is simple and inevitable yet profound and evidently far-reaching. This no doubt is another instance of universality of the fluctuation-dissipation theorem.

References

[1] Higgins et al. Physiological and pathological population dynamics of circulating human red blood cells. Proc Natl Acad Sci U S A. 2010 Nov 23;107(47):20587-92. Epub 2010 Nov 8.