Fractal Networks Explain Regional Myocardial Flow Heterogeneity

Abstract

Regional myocardial blood flow is very heterogeneous. This has been found by injection of radioactively labeled microspheres, or the “molecular microsphere” iododesmethylimipramine, and measuring the deposition of these flow indicators in small sample pieces into which the heart has been cut (King et al., 1985; Bassingthwaighte et al., 1987; Bassingthwaighte et al., 1988). It was also found that the spread of the flow distribution increases with the spatial resolution of the measurement. This could be expressed (Bassingthwaighte et al, 1988; Bassingthwaighte, 1988) via a mathematical relation between the relative dispersion RD, defined as the standard deviation of the flow distribution divided by its mean, and the average mass, m, of the sample pieces into which the heart was divided:

$${\rm{RD}}\left( {\rm{m}} \right){\rm{ = RD}}\left( {{{\rm{m}}_{\rm{0}}}} \right) \cdot {\left[ {{{\rm{m}} \over {{{\rm{m}}_{\rm{0}}}}}} \right]^{{\rm{1 - D}}}}$$

(1)

where m_o is an arbitrary reference mass. Since the number of sample pieces equals the total mass M divided by the average mass of the sample piece it follows that

$${\rm{RD}}\left( {\rm{N}} \right){\rm{ = RN}}\left( {{{\rm{N}}_{\rm{0}}}} \right) \cdot {\left[ {{{\rm{N}} \over {{{\rm{N}}_{\rm{0}}}}}} \right]^{{\rm{D - 1}}}}$$

(2)

The power laws fit the measurements very well for all except the larger sample pieces (see Figure 1). Such a relation between a measure and the spatial resolution of the measurement has been found for the geometrical features of certain types of mathematical sets that are called fractals. Equations 1 and 2 are therefore often called fractal relationships and parameter D is called the fractal dimension.