# Zin Lin

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.

## 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. 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

To begin with, it is assumed that the reduction in RBC volume and Hb content follows a Brownian-like behavior described 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>:

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>:

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

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 lower than that of healthy and ACD whereas <math>\alpha</math> of TT and IDA 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].