Types of vascular resistance depending on the network
The human circulatory system is a complex hydrodynamic network. Total resistance within it is formed based on how the vessels connect to one another.
- Series connection. This type is characteristic of sections of the vascular bed where vessels transition into one another (e.g., a large artery transitions into a smaller one). Total resistance ($R$) in such a section is calculated extremely simply: it is the sum of the resistances of each individual segment. The formula is: $R = R_1 + R_2 + R_3$.
- Parallel connection. Observed in a branched vascular network. Here, Kirchhoff's laws for parallel pipes apply. With such branching, it is not the resistances themselves that are summed, but the conductances ($C$). Conductance is the reciprocal of resistance ($C = 1/R$). Accordingly, the formula takes the form: $C = C_1 + C_2 + C_3$.
From this follows a most important hemodynamic rule: the total resistance of parallel vessels is always less than the resistance of each individual vessel.
Hagen-Poiseuille equation
To calculate the hydrodynamic resistance to blood flow in a specific section of the network, physiologists use the classical Hagen-Poiseuille equation (1846).
The calculation formula is as follows: $R = \frac{8l\eta}{\pi r^4}$
This formula encodes the main factors affecting hydrodynamics:
- Blood viscosity ($\eta$). Has a direct relationship with resistance. The higher the viscosity, the harder it is for blood to flow, and the higher the resistance.
- Vessel length ($l$). Also has a direct relationship. The longer the tube, the higher the fluid resistance.
- Vessel radius ($r$). This is the most critical parameter, having an inverse relationship. Resistance is inversely proportional to the vessel radius to the fourth power ($r^4$).
The formula also uses the constant $\pi$ — the ratio of the vessel's circumference to its diameter.
Two fundamental physiological dependencies follow from the Poiseuille equation. First, the volumetric blood flow rate is strictly proportional to the vessel radius. Second, due to the fourth power of the radius, even the most insignificant change in vessel lumen causes a sharp, avalanche-like change in hydrodynamic resistance.
Distribution of resistance and energy in the vascular bed
Different parts of the circulatory system make unequal contributions to the total resistance to blood flow.
- Arterioles. They are the main resistance vessels. They exert the maximum resistance to blood movement. It is at this level of the vascular bed that the most significant drop in blood pressure occurs.
- Capillaries. They offer slightly less resistance than arterioles. At first glance, this seems illogical since their radius is smaller. However, the reason lies in length: capillaries are very short ($l$) compared to arterioles, which compensates for the narrowness of the lumen.
This distribution of resistance directly affects myocardial function. The energy expenditure of the heart for propelling blood through vessels is distributed as follows:
- 85% of energy is spent overcoming resistance in arterioles and capillaries;
- 10% is expended in large arteries;
- 5% is sufficient to propel blood through veins.
Briefly: answers to topical questions
By what formula is total peripheral resistance (TPR) calculated via blood pressure and cardiac output?
The formula for calculating total peripheral resistance (TPR) is directly based on the main equation of circulatory regulation.
According to this rule, blood pressure is the product of two parameters:
- Basic formula — BP = cardiac output (CO) × total peripheral resistance (TPR).
It follows from this equation that TPR can be expressed as the ratio of blood pressure to cardiac output. The regulation of these parameters itself is carried out through four key points: arterioles (responsible for resistance), venules, the heart, and the kidneys.
What physiological parameters directly determine blood viscosity?
Blood viscosity directly depends on the number of formed elements, the presence of specific plasma proteins, and other physiological parameters.
The main factors determining viscosity include:
- Number of formed elements — primarily erythrocytes.
- Presence of plasma proteins — determine rheological properties.
- Hematocrit — has a direct relationship (the higher it is, the higher the viscosity).
- Acid-base balance (ABB) — has an inverse relationship.
- Hypercapnia — increases viscosity (therefore, venous blood viscosity is higher than arterial).
- Blood flow velocity — in microvessels, viscosity substantially depends on this parameter.
What is the role of precapillary sphincters in shaping vascular bed resistance?
The role of precapillary sphincters is to regulate the number of perfused (functioning) capillaries at any given moment in time.
They are classified as resistance vessels and have the following features:
- Basal myogenic tone — controlled by tissue metabolites, which predominantly exert a vasodilatory effect.
- Regulation mechanism — contraction of smooth muscle cells blocks the vessel lumen, which directly limits blood flow into the capillary bed and shapes local resistance to blood flow.
4th power rule: if a vessel narrows by just 2 times, its resistance to blood flow will increase by 16 times (2 to the 4th power).
Class and exam questions
Why do capillaries, being the narrowest vessels, offer lower resistance than arterioles?
It is all about Poiseuille's equation, which takes into account not only the radius but also the length of the vessel. Despite their extremely small radius, capillaries have a very short length compared to arterioles, so their total hydrodynamic resistance turns out to be lower.
How is resistance calculated in a branched network (e.g., in the capillary bed)?
The branched vascular network represents a parallel connection. In this case, conductances are summed rather than resistances ($C = 1/R$). Therefore, the total resistance of the entire network will always be less than the resistance of a single individual vessel within it.
What will happen to resistance if blood viscosity increases?
According to the Hagen-Poiseuille equation, there is a direct relationship between blood viscosity and hydrodynamic resistance. Consequently, with an increase in viscosity, resistance to blood flow will increase.
What else to study on this topic
- Mathematical derivation of the Hagen-Poiseuille equation
- Application of Kirchhoff's laws to hemodynamics
- Mechanisms of blood viscosity changes in the body
- Physiological role of arterioles as the taps of the vascular bed
- Changes in volumetric blood flow rate during vasoconstriction
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