The S Factor—A New Derived Hemodynamic Oxygenation Parameter—A Useful Tool for Simplified Mathematical Modeling of Global Problems of Oxygen Transport

Abstract

We describe a new derived hemodynamic oxygenation parameter, the S factor (S). The factor is based on oxygen delivery and oxygen consumption and can range from -3 to 1. It allows simplified mathematical modeling of clinical problems of oxygen transport and can be applied to many clinical situations.

A new hemodynamic oxygenation parameter, the S factor (S), is introduced as an aid to mathematical modeling. It is defined as follows:

$$\text{S}= \frac{\dot{\text{D}}\text{O}_{2}-4\dot{\text{V}}\text{O}_{2}}{\dot{\text{D}}\text{O}_{2}}$$

$\left ( \text{S}= {\text{oxygen delivery}, \dot{\text{V}}\text{O}_{2}} =\text{oxygen consumption} \right )$ S can theoretically vary from $S = \frac{{\dot D{O_2} - 4\dot V{O_2}}}{{D{O_2}}}$ When $\dot D{O_2}/\dot V{O_2} = 4$ An S < 0 implies utilization of reserve oxygen transport capacity. An S > 0 implies increased oxygen delivery in relation to oxygen consumption (ie. “shunted oxygen delivery”). By algebraic manipulation and substitution of the components of $\dot{\text{D}}{\text{O}}_{2}$ into Equation 1:

$$\dot{\text{D}}{\text{O}}_{2}= \dot{\text{Q}} \times {\text{Ca}} \times {10}$$

$$\dot V{O_2} = \dot Q\left( {Ca - Cv} \right){\text{ x }}10$$

(1)

the following equations can be derived:

$$\dot{\text{Q}} = \frac{4\dot{\text{VO}_{2}}}{[(1-\text{S})[1.36\text{Hb} \text{(Sat)} =(.0031\text{PaO}_{2})]10}$$

$${\text{Hb}} = \frac{\dot{\text{V}}\text{O}_{2}}{(1-\text{S})[1.36\text{Hb}(\text{Sat}) = (.0031 \text{PaO}_{2})]10}$$

Ca — Cv (Ca = arterial content, Cv = venous content) can be determined by substituting components of oxygen consumption:

$$\dot{\text{V}}{\text{O}}_{2}= \dot{\text{Q}} (\text{Ca}-\text{Cv}) \times 10$$

into equation 1 and solving for Ca — Cv.

$$(\text{Ca}-\text{Cv})= \frac{\dot{\text{D}}\text{O}_{2}- (1-\text{S})}{4\dot{\text{Q}} \times 10}$$

Equation 6 can be simplified to:

$$(\text{Ca}-\text{Cv})= \frac{{\text{Ca}}(1-\text{S})}{4}$$

A previously defined relationship¹ between mixed venous PO₂ (PvO₂) and $\dot D{O_2}/\dot V{O_2}$ (where calculated P₅₀ is 26.6 ± 1.0) can be used to modify S in a clinically relevant manner.

$$Pv{O_2} = 5.44\dot D{\text{ }}{{\text{O}}_2}/\dot V{O_2} + 18.16$$

((8))

The relationship between S and PvO₂ can be defined by substituting Equation 4 into Equation 1 and solving for PvO₂

$$Pv{O_2} = \left[ {21.76/\left( {1 - S} \right)} \right] + 18.16$$

((9))

As an example, at a PvO₂ of 28 torr (anaerobic threshold), S = −1.2. The relationship between PvO₂ and S is shown in Figure 1. S, which can also be defined as $1-4(\dot{\text{V}}\text{O}_{2}/\dot{\text{D}}\text{O}_{2})$ or 1 — 4(OER), is a useful tool for mathematical modeling of global problems of oxygen transport because the previously derived equations with the S value allow the components of oxygen transport to be interrelated in a clinically relevant manner. Additional advantages of using S in mathematical modeling are:

• 1. Conceptually it ‘fits’ in that in regards to the sign (+ or -), as a -S implies utilization of reserve oxygen transport capacity and a +S implies wasted or excess oxygen delivery (shunted).

• 2. These concepts are easily quantified using the S factor.

• 3. It’ spreads out’ the difference between values for parameters (OER or S) integrating components of oxygen transport, ie. in the ‘normal state’ regarding oxygen transport, OER = 0.25 and S = 0. At the anaerobic threshhold (PvO₂ = 28 torr), OER = 0.55 and S = -1.2. Thus, the change in OER from ‘normal state’ to anaerobic threshold is 0.3 (0.55 — 0.25) and the change in S is 1.2. This represents a four-fold increase.

Four examples of mathematical modeling of global problems of oxygen transport using the S factor are described below.