NOTES : A Model of Gas Exchange for Hyperpolarized Xe(129) Magnetic Resonance of the Lung
背景知识：
Gas exchange is the essential function of the lung. In general, a lung can be viewed as a porous medium（多孔介质） consisting of capillary（毛细管） circuits with blood flowing inside. The blood flows are separated from the air spaces by several layers of tissues, including epithelium（组织上皮细胞）, endothelium（内皮）,and interstitium（小间隙）.These tissues together are called air–blood barrier.（空气血液栅栏）.At equilibrium, gas molecules in the alveolar spaces（肺泡空间） are under constant exchange with those dissolved in the barrier and capillary blood.
Remark: 气体由肺泡扩散浓度自高处向低处，平衡状态时，以常数速率和溶解在组织和毛细管血管的气体交换.
Almost all pulmonary diseases can be attributed to deficient gas exchange or delivery in the lung.
气体交换不足或者过量均会导致肺部疾病，联系5个肺部关键参数。
Prob: How measure gas exchange？
Ans: However, none of the established imaging techniques provides direct measurement of gas exchange！！！
没有直接的方法，间接实验。
Exp.1 computerized tomography （ 断层摄影技术 ） measures tissue density
Exp.2 magnetic resonance imaging of hyperpolarized (HP)
He (超极化氦气) images the air spaces in the lung.
Exp.3 MR of HP Xe is capable of providing direct measurements of gas exchange in the lung. As a contrast
agent, not only does xenon yield dramatically enhanced
MR signals in the air spaces, but it also dissolves into lung tissue and blood. As the dissolved xenon in blood follows the same physiological pathways of the normal blood gases (i.e., O2,CO2) (参看下面的图)
量化：
Quantification of dissolved xenon dynamics will lead to a quantified understanding of lung function.
(How? M.R. experiment)
原理：
A particular feature of xenon that permits such a study, is that the xenon dissolved into human lung exhibits two large chemical shifts from the resonance frequency（共振频率） of the free xenon gas—one at 197 ppm（parts per million）, for xenon in lung tissue and blood plasma（血浆） (TP xenon), the other at 217 ppm, for the xenon in the red blood cells (RBC xenon)
RBC ：Red Blood Cell
TP : Tissue and Plasma（组织和血浆）
Xenon ----> Dissolved Xenon +Free Xenon
Dissolved Xenon = TP Xe + PBC Xe
197ppm: xenon in lung tissue and blood plasma（血浆）
217ppm: the xenon in the red blood cells
核磁共振具体实验技术暂不知。
CSSR has been used frequently, over the past years in various lung diseases, including fibrosis（纤维变性） and emphysema （肺气肿）.
Despite the previous applications of CSSR, there has not been a satisfactory theory to interpret xenon uptake dynamics for both dissolved xenon peaks in the lung.
density M_d within the septum（隔膜）, perpendicular（垂直的） to the blood flow
M_d is density of dissolved xenon
M_f is the density of free xenon gas (at 0 ppm) in the air spaces
D is the diffusion coefficient of dissolved xenon
\lambda is the Ostwald solubility of xenon in lung parenchyma（软细胞组织）
Prob: How to define the boundary value conditions？
Apply Fourier’s separation of variable obtain series solution:
Where T is the xenon-exchange time constant in the lung
(That is Par.4)
Proof:
。。。。。。。。
初边值问题的数值解（张文生book）：
级数解渐进:
Assume:
S_A is the total surface area of air space
V_g is total volume of the air spaces in the lung
The dissolved xenon signal is proportional to M_d *S_A/2
The free xenon gas signal is proportional to M_f * V_g
The normalized signal distribution S_d(x, t) for dissolved xenon can be written as (WHY? Normalized)
和 (4)式进行对比 compare with (4).
S_d= M_d SVR / 2M_f
S_A / V_g is the surface-area-to-volume ratio
(That is Par.1 SVR)
S_d1(t): The Total Signal From The Tissue
（组织中xenon 信号）
It is not difficult to derive (6)--->(7),.calculated as the spatial integral of S_d in Eq.6 over the two regions from (0) to (\delta) and from (d–\delta) to (d).
Where，b is the normalization factor (dimensionless), and \delta/d is what we call barrier-to-septum ratio.
(That is Par.2 BSR)
The dissolved xenon signal from the blood (‘‘blood xenon’’) is more difficult to calculate owing to flow.
S_d2(s): The Total Signal From Blood (No-flow & flow )
(总血液中Xenon 信号)
No flow : integral of S_d at (\delta ,d-\delta)
How is consider blood flow? (Key)
The total xenon signal from the blood is：
Where t_X is pulmonary capillary transit time
(Par.5 运输时间)
To calculate the xenon signal in tissue and blood plasma, e.g., TP xenon signal S_TP(t) at 197 ppm, and signal of RBC xenon S_RBC(t) at 217 ppm, we let h denote the fraction of RBC xenon relative to total xenon in blood
S_PBC = h * S_d2
S_blood plasma =(1-h) * S_d2
S_TP = S_d1 + (1-h) * S_d2（组织和血浆信号）
The hematocrit (HCT) can be calculated from h using
(Par.3 血细胞比容)
Where \lambda_RBC and \lambda_P are Ostwald solubilities of xenon in RBCs and plasma, respectively.
Remark : How calculate h ? MR of HP Xe
上面建立的理论模型，实验信号可观测到（不同的肺部疾病），接下来拟合需要的五个参数与正常的对比下？？？
文章选取了肺部处于四种不同的状态。
Remark:
COPD (Chronic Obstructive Pulmonary Disease)慢性阻塞肺病
Fibrosis 纤维病变
Anemia 贫血
正常健康人的参数：
\lambda 0.2
\delta 2
d 10
SVR 250
T 40
\eta 0.5
t_X 1.6