Development and Application of a Logistic-Based Systolic Model for Hemodynamic Measurements Using the Esophageal Doppler Monitor

Abstract

The esophageal Doppler monitor (EDM) is a clinically useful device for minimally invasive assessment of cardiac output, preload, afterload, and contractility. An empirical model, based upon the logistic function, has been developed. Use of this model illustrates how the EDM could estimate the net effect of aortic and non-aortic contributions to inertia, resistance, and elastance within real time. This is based on an assumed mechanical impedance conceptually resembling that of a series arrangement of a spring, mass, and dashpot. In addition, when used with an invasive radial arterial catheter, the EDM may also estimate aortic pulse wave velocity, as well as aortic characteristic impedance, and characteristic volume. Approximations of left ventricular stroke work and stroke power can also be made. Furthermore, the effects of inertia, resistance, and elastance, on mean blood pressure during systole, can be quantified. These additional parameters could offer insight for clinicians, as well as researchers, and may be beneficial in further examining and utilizing clinical hemodynamics with the EDM. These additional measurements also underscore the need to integrate the EDM with existing and future monitoring equipment.

Appendices

Appendix A

Determining Cardiac Output Using EDM Parameters (Boulnois and Pechoux 2000)

Stroke distance, in the distal aorta, SD_aorta, is determined from the integral of distal aortic blood flow velocity over flow time:

$$\hbox{SD}_{\rm aorta} =\int\limits_{0}^{\rm FT}v(t){\rm d}t.$$

(A.1)

Note that the average velocity, of distal aortic blood flow $\bar{V}$, is:

$$ \bar{V}=\frac{1}{\hbox{FT}}\int_{0}^{\rm FT}v(t)\hbox{d}t.$$

(A.2)

Therefore, SD_aorta is equivalent to the product of average velocity and flow time:

$$ \hbox{SD}_{\rm aorta} = (\bar{V})(\hbox{FT}). $$

(A.3)

Stroke volume in the distal aorta is:

$$ \hbox{SV}_{\rm aorta} = (\hbox{SD}_{\rm aorta})(A). $$

(A.4)

where A is the distal aortic cross-sectional area. Thus:

$$ \hbox{SV}_{\rm aorta} = (\bar{V})(A)(\hbox{FT}).$$

(A.5)

That portion of cardiac output, which flows through the distal aorta, is then:

$$ \hbox{CO}_{\rm aorta} = (\hbox{SV}_{\rm aorta})(\hbox{HR}) = (\bar{V}) (A)(\hbox{FT})(\hbox{HR}). $$

(A.6)

In this application, flow time, FT, has units of seconds/beat and heart rate, HR, has units of beats/second. Therefore the product: FT × HR is dimensionless.

Total cardiac output is then:

$$ \hbox{CO} = (\bar{V})(A)(\hbox{FT})(\hbox{HR})(1.4). $$

(A.7)

where 1.4 is a dimensionless constant which is based upon a linear regression analysis from clinical data (Boulnois and Pechoux 2000).

If distal aortic cross-sectional area is unknown, minute distance within the aorta, MD_aorta, can be defined as:

$$ \hbox{MD}_{\rm aorta} = (\bar{V})(\hbox{FT})(\hbox{HR}).$$

(A.8)

Therefore, MD_aorta = SD_aorta · HR. Thus, MD_aorta correlates with total cardiac output.

Appendix B

Determining Stroke Distance, in the Distal Aorta, by Integrating Aortic Blood Flow Velocity Over Time

Using the logistic-based systolic model:

$$ {sd}(t)=\int{v(t){\rm d}}t=\int\alpha\beta \hbox{e}^{-\gamma t}\left[\left(1-\frac{t}{\hbox{FT}}\right) t\right]\hbox{d}t $$

(B.1)

Separating the above so that sd(t) =  I ₁ − I ₂—constant of integration:

$$I_{1} =\int\alpha\beta \hbox{e}^{-\gamma {t}}t\,{\rm d}t = \frac{(\gamma t\hbox{e}^{-\gamma t} -\hbox{e}^{-\gamma {t}})}{\gamma^2}\alpha\beta $$

(B.2)

$$ I_{1} =\frac{-(\gamma t+1)}{\gamma^2}\alpha\beta \hbox{e}^{-\gamma {t}} $$

(B.3)

$$ I_{2}=\int \alpha\beta \hbox{e}^{-\gamma {t}}\frac{t^2} {\hbox{FT}}\,\hbox{d}t $$

(B.4)

$$ I_{2} =\frac{-(\gamma^2 t^2\hbox{e}^{-\gamma t}+ 2\gamma t\hbox{e}^{-\gamma t}+2\hbox{e}^{-\gamma t})}{\gamma^3}\frac{\alpha\beta}{\hbox{FT}} $$

(B.5)

$$ I_{2} =\frac{(-(\gamma t)^2-2\gamma t- 2)}{\gamma^3}\frac{\alpha\beta}{\hbox{FT}}\hbox{e}^{-\gamma t} $$

(B.6)

$$ I_{1} - I_{2} =\left(\frac{-(\gamma t+1)}{\gamma^2}- \frac{(-(\gamma t)^2-2\gamma t- 2)}{\gamma^3\hbox{FT}} \right)\alpha\beta \hbox{e}^{-\gamma t} $$

(B.7)

$$ I_{1} - I_{2} =\frac{(-\gamma^2 t\hbox{FT}-\gamma \hbox{FT}+(\gamma t)^2+2\gamma t+2)}{\gamma^3\hbox{FT}}\alpha\beta \hbox{e}^{-\gamma t} $$

(B.8)

$$ I_{1} - I_{2} =\left(\frac{- t}{\gamma}-\frac{1}{\gamma^2} +\frac{t^2}{\gamma \hbox{FT}}+\frac{2 t}{\gamma^2 \hbox{FT}}+\frac{2}{\gamma^3\hbox{FT}}\right)\alpha\beta \hbox{e}^{-\gamma t} $$

(B.9)

$$ I_{1} - I_{2} =\left(\frac{\left(-1+\frac{1}{\hbox{FT}}\right)t}{\gamma} +\frac{\left(-1+\frac{2 t}{\hbox{FT}}\right)}{\gamma^2}+ \frac{2}{\gamma^3\hbox{FT}}\right)\alpha\beta \hbox{e}^{-\gamma t} $$

(B.10)

The constant of integration is then chosen so that sd(0) = 0:

$$ {sd}(t) = \left(\frac{\left(-1+\frac{1}{\hbox{FT}}\right)t}{\gamma} +\frac{\left(-1+\frac{2 t}{\hbox{FT}}\right)}{\gamma^2}+ \frac{2}{\gamma^3\hbox{FT}}\right)\alpha\beta \hbox{e}^{-\gamma t}-\left(-1+\frac{2}{\gamma \hbox{FT}}\right)\frac{\alpha\beta}{\gamma^2}.$$

(B.11)

Thus:

$$ \int v(t)\hbox{d}t={sd}(t)=\left[\left[\left(\frac{t}{\hbox{FT}}-1 \right)t+\left(2\frac{t}{ \hbox{FT}}-1\right)\frac{1}{\gamma}+\frac{2}{\hbox{FT}\gamma^2}\right] \frac{\alpha\beta}{\gamma}\hbox{e}^{-\gamma t}\right]-\left(-1+\frac{2}{\gamma \hbox{FT}}\right)\frac{\alpha\beta}{\gamma^2}. $$

(B.12)

Figure A1 is plot of sd(t) which reveals the familiar sigmoid curve that is characteristic of the logistic function.

Fig. A1

A plot of stroke distance, sd(t), as a function of time. Note its sigmoid shape which is characteristic of the logistic function

Evaluating ${sd}(t)\vert^{\rm FT}_{0}$ yields SD_aorta = sd(FT) since sd(0) = 0:

$$ \hbox{SD}_{\rm aorta} = {sd(\rm FT)} - {{sd}(0)} =\left[\left( 1+\frac{2}{\hbox{FT}\gamma}\right)\hbox{e}^{-\gamma {\rm FT}}-\left(-1+\frac{2}{\hbox{FT}\gamma}\right)\right]\frac{\alpha\beta}{\gamma^2}. $$

(B.13)

Appendix C

Derivation of the Bramwell–Hill Equation (Bramwell and Hill 1922)

Tension, T, within the wall of a compliant cylinder can be described as:

$$ T = \sigma h = r\Updelta \hbox{P}. $$

(C.1)

where σ is stress and h is wall thickness. It is assumed that h is small compared to radius, r. Furthermore, both internal and external radii are approximately equal and are represented as the constant r. ΔP represents the difference between external and internal wall pressures.

Therefore, wall stress is (Noordergraaf 1978):

$$ \sigma=\frac{r\Updelta P}{h}.$$

(C.2)

Strain, ɛ, is defined as:

$$ \varepsilon=\frac{\Updelta r}{r}. $$

(C.3)

Young’s modulus, E, is:

$$ E =\frac{\sigma}{\varepsilon} =\frac{\frac{r\Updelta P}{h}} {\frac{\Updelta r}{r}}=\frac{r^2\Updelta P}{h\Updelta r}. $$

(C.4)

The change in radius, due to the change in pressure, can then be represented as:

$$ \Updelta r=\frac{r^2\Updelta P}{hE}. $$

(C.5)

Compliance is defined as the change in volume divided by the change in pressure:

$$ C=\frac{\Updelta \hbox{Vol}}{\Updelta P}=\frac{\pi r_{\rm f}^2 L_{\rm e}-\pi r_{\rm i}^2 L_{\rm e}}{\Updelta P}= \frac{\pi L_{\rm e}(r_{\rm f}^2- r_{\rm i}^2)}{\Updelta P}. $$

(C.6)

where r _f is the final radius and r _i is the initial radius and L _e is the length of the compliant cylinder. Recognizing that (r ²_f −r ²_i ) is the difference of two squares, (C.6) can then be expressed as:

$$ C=\frac{\pi L_{\rm e}(r_{\rm f}+ r_{\rm i})(r_{\rm f}- r_{\rm i})}{\Updelta P}. $$

(C.7)

Noting that $r_{\rm f}+ r_{\rm i}\approx 2 r$ and r _f−r _i = Δr, then (C.7) can be expressed as:

$$ C=\frac{\pi L_{\rm e}(2 r)(\Updelta r)}{\Updelta P}.$$

(C.8)

Substituting (C.5) into (C.8) yields:

$$ C=\frac{\pi L_{\rm e}(2 r)(r^2\Updelta P)}{hE\Updelta P}. $$

(C.9)

Compliance can then be expressed as:

$$ C=\frac{2 \cdot \hbox{Vol} \cdot r}{hE}.$$

(C.10)

Young’s modulus can then be represented as:

$$ E=\frac{2\cdot \hbox{Vol}\cdot r}{hC}. $$

(C.11)

The Moens–Korteweg equation (Milnor 1989) relates pulse wave velocity, in a compliant cylinder, to its geometric and physical properties:

$$ v_{pw}=\sqrt{\frac{Eh}{2\rho r}}. $$

(C.12)

Substituting (C.11) into (C.12) and simplifying yields the Bramwell–Hill equation ([Milnor 1989; Liu et al. 1989):

$$ v_{pw}=\sqrt{\frac{\frac{2\cdot \hbox{Vol} \cdot r}{hC} h}{2\rho r}}=\sqrt{\frac{\hbox{Vol}}{\rho C}}. $$

(C.13)