Agilo L. Kern^{1,2}, Karen M. Olsson^{2,3}, Till F. Kaireit^{1,2}, Frank K. Wacker^{1,2}, Jens M. Hohlfeld^{2,3,4}, Marius M. Hoeper^{2,3}, and Jens Vogel-Claussen^{1,2}

^{1}Institute of Diagnostic and Interventional Radiology, Hannover Medical School, Hannover, Germany, ^{2}Biomedical Research in Endstage and Obstructive Lung Disease Hannover (BREATH), German Center for Lung Research (DZL), Hannover, Germany, ^{3}Department of Respiratory Medicine, Hannover Medical School, Hannover, Germany, ^{4}Clinical Airway Research, Fraunhofer Institute for Toxicology and Experimental Medicine, Hannover, Germany

The hyperpolarized ^{129}Xe chemical shift saturation recovery
(CSSR) method combined with mathematical models of gas uptake provides unique
insights into lung microstructure and function. Here, we propose a
generalization of the Patz model for assessing interstitial membrane permeability. We test the model’s
potential for in-vivo determination of membrane permeability by fitting to CSSR
data from healthy volunteers and pulmonary hypertension (PH) patients. The obtained permeabilities lie between those estimated for O_{2}
and CO_{2} in healthy volunteers and tend to be lower in PH. The proposed model enables the
non-invasive assessment of interstitial barrier permeability and shows
potential as diagnostic tool.

$$\frac{\partial}{\partial t}m(x,t)=D_b\frac{\partial^2}{\partial x^2}m(x,t)$$

subject to the boundary conditions

$$D_b\frac{\partial}{\partial x}m(x,t)\rvert_{x=0}=\kappa(m(0,t)-\lambda m_0)$$

$$D_b\frac{\partial}{\partial x}m(x,t)\rvert_{x=L}=-\kappa(m(L,t)-\lambda m_0)$$

$$m(x,0)=0,$$

where $$$D_b=1.35\cdot 10^{-9}\mathrm{m}^2/\mathrm{s}$$$ denotes the

This problem is mathematically equivalent to heat conduction in a rod subject to convection boundary conditions and has been studied in detail previously.

$$m(x,t)=\lambda m_0+\sum_{n=1}^\infty c_n\left[\cos\left(\frac{\mu_nx}{L}\right)+\frac{B}{\mu_n}\sin\left(\frac{\mu_nx}{L}\right)\right]\exp\left(-\frac{D_b\mu_n^2}{L^2}t\right)$$

with

$$c_n=-\lambda m_0\frac{\int_0^L\mathrm{d}x\left[\cos\left(\frac{\mu_n x}{L}\right)+\frac{B}{\mu_n}\sin\left(\frac{\mu_n x}{L}\right)\right]}{\int_0^L\mathrm{d}x\left[\cos\left(\frac{\mu_n x}{L}\right)+\frac{B}{\mu_n}\sin\left(\frac{\mu_n x}{L}\right)\right]^2}=-2B\lambda m_0\frac{(-1)^{n+1}+1}{B^2+\mu_n^2+2B}$$

and the $$$\mu_n$$$ solve the transcendental equation

$$\cot(\mu_n)=\frac{1}{2}\left(\frac{\mu_n}{B}-\frac{B}{\mu_n}\right)$$

with the diffusional Biot number $$$B=\frac{\kappa L}{D_b}$$$.

Integrating the magnetization in the septum and treating blood flow as in

$$F(t)=\frac{\lambda L}{2}\eta\frac{S_a}{V_g}\frac{\tau-t}{\tau}\left[1-\sum_{n=1}^{\infty}\frac{(-1)^{n+1}+1}{B^2+\mu_n^2+2B}\frac{4B^2}{\mu_n^2}\exp\left(-\frac{D_b\mu_n^2}{L^2}t\right)\right]+\lambda L\eta\frac{S_a}{V_g}\frac{1}{\tau}\left[t+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}+1}{B^2+\mu_n^2+2B}\frac{4B^2L^2}{D_b\mu_n^4}\left(\exp\left(-\frac{D_b\mu_n^2}{L^2}t\right)-1\right)\right].$$

with $$$\eta$$$ the RBC fraction of

Since the function $$$F(t)$$$ depends on the solutions $$$\mu_n$$$ of a transcendental equation, it cannot be written in closed form. Fitting was thus performed using the unconstrained nonlinear optimization algorithm fminsearch of Matlab (R2014b, MathWorks, Ismaning, Germany). Only the RBC uptake data was used for fitting $$$F(t)$$$ to ensure that the signal originates solely from blood. The surface-to-volume ratio was used as external parameter obtained by fitting the well-known Butler formula to the sum of RBC and tissue/plasma (TP) data and assuming $$$D_t=3.3\cdot 10^{-10}\mathrm{m}^2/\mathrm{s}$$$

In addition, RBC amplitude oscillations from dynamic spectroscopy were determined similarly as previously described.

Membrane permeability, capillary transit time and RBC fraction tended to be lower in PH patients compared to healthy volunteers. The amplitude of RBC signal oscillations tended to be lower in patients of idiopathic and hereditary pulmonary arterial hypertension compared to those with secondary PH.

Although the proposed model in principle allows for separate determination of $$$\kappa$$$ and $$$L$$$, we found that in the presence of realistic noise levels one can only determine one of both parameters with the other one fixed. The assumed $$$L$$$ is roughly the size of both a normal capillary and an individual RBC. While in real lung tissue capillaries may collapse,

While the permeability of the interstitial membrane to

It was previously described that RBC oscillation amplitudes differ between subtypes of PH due to increased impedance of arterioles in pulmonary arterial hypertension.

1. Sta Maria N, Eckmann DM: Model Predictions of Gas Embolism Growth and Reabsorption during Xenon Anesthesia. Anesthesiology 2003;99:638–64.

2. West JB. Respiratory Physiology: The Essentials. Lippincott Williams & Wilkins 2008. ISBN 978-0-781-77206-8.

3. Ladefoged J, Andersen AM: Solubility of Xenon-133 at 37°C in Water, Saline, Olive Oil, Liquid Paraffin, Solutions of Albumin, and Blood. Phys Med Biol 1967;12:353–358.

4. Hahn DW, Özisik MN. Heat Conduction. John Wiley & Sons 2012. ISBN 978-1-118-33011-1.

5. Patz S, Muradyan I, Hrovat MI et
al. Diffusion of hyperpolarized ^{129}Xe in the lung: a simplified
model of ^{129}Xe septal uptake and experimental results. New J Phys
2011;13:015009.

6. Butler JP, Mair RW, Hoffmann D et al. Measuring surface-area-to-volume ratios in soft porous materials using laser-polarized xenon interphase exchange nuclear magnetic resonance. J Phys-Condens Mat 2002;14:L297–L304.

7. Ruppert K, Mata JF,
Brookeman JF, Hagspiel KD, Mugler JP III. Exploring
Lung Function With Hyperpolarized ^{129}Xe Nuclear Magnetic Resonance. Magn Reson
Med 2004;51:676–687.

8.
Wang
Z, Bier EA, Swaminathan A et al. Diverse Cardiopulmonary Diseases are
Associated with Distinct Xenon MRI Signatures. Eur Respir J 2019; in press.

9. Weibel ER, Sapoval B, Filoche M. Design of peripheral airways for efficient gas exchange. Resp Physiol Neurobi 2005;148:3–21.