Linear decay of red blood cell populations

How can we use linear mathematical models to calculate population growth or decay in Red Blood Cells?

Simple mathematical models of population extinction cannot accurately describe the decline of some populations, such as red blood cells (RBCs). Due to the varying biology of cell systems, simple linear decay assists with modelling population decay of RBCs, as death is not uniform, exponential decay models are often not accurate for these scenarios.  

 Linear decay happens when a quantity decreases by the same amount at regular intervals. For example, the annual depreciation of a car and the decomposition of a pollutant in the air by a negative constant amount. Linear decay can be described by the rule for the nth term such that: 

 tn = t1 + (n-1)d 

Where: 

t1 = the first term 

n = the term number, and 

d = the common difference between the terms in a sequence 

 Exponential decay happens when a quantity decreases at a rate proportional to its current value at regular intervals. For example, the decreasing height of a ball at each bounce. 

 In the case of RBC extinction, linear decay could be justified because cell death is not uniform as cells biologically behave differently (population heterogeneity). If cell behaviour was the same (homogeneity), then an exponential decay calculation would be more appropriate, as deaths would be proportional and continually increasing.  

A sub-exponential model of population extinction can be used when a population exhibits a linear pattern of decay using the following differential equation: 

dN/dt = – KNP. 

Where  

t = time 

N(0) =  the initial population size 

k =  the per capita death rate, and 

P =  the typical exponential death. 

The results of this model will show that RBCs with individual death rate parameters will die at a rate similar to their frequency in the population. It will exhibit RBCs’ linear pattern of decay and can be used to describe the effects of medicinal drugs that may cause RBC depletion.   

Like RBCs, other biological systems may be subject to linear decay and this proposed model can capture their linear pattern of decay and should be considered when developing population models.   

Adapted from: Kareva, I., & Karev, G. (2020). Linear rather than exponential decay: a mathematical model and underlying mechanisms. arXiv preprint arXiv:2004.05726.   

Read the original article 

Questions for students: 

1. Define decay.
2. Define growth. 
3. Explain the difference between linear decay and exponential decay. 
4. Draw a diagram that shows the differences between a linear decay and an exponential decay. 

Key words for students to understand: 

• Mathematical models 
• Population extinction 
• Linear decay 
• Exponential decay 
• Depreciation 
• Decomposition 
• Proportional 
• Extinction