**2.1.3 Linear coupled system of the loop of Henle and analytical solutions**

In this model, the driving force for increasing the osmolality (largely contributed by the concentration of Na+) of the descending limb is proportional to the difference in the osmotic gradient between it and renal interstitium. The renal interstitium itself has a osmotic concentration proportional to the ascending limb of the loop of Henle, due to active transport of Na+ out of the ascending limb.

The osmolality (largely due to the concentration of Na+) of the ascending limb is modelled on active sodium transport and hence the rate of fall is only related to its own concentration/osmolality.

The concentration of the interstitium is largely identical to the concentration in the descending tubule as there is passive movement of water through the descending tubule. The schematic figure 4 illustrates the model of the renal tubule.

Fig. 4. Schematic diagram of the loop of Henle.

*x* the distance along the loop of Henle measured from the origin of the descending limb (in mm)

*L* the total actual length of the loop of Henle measured along one of the limb (in mm)

*Cd* the concentration of Na+ in the descending limb (in mOsm/L)

*Ca* the concentration of Na+ in the ascending limb (in mOsm/L)

*Ci* the concentration of Na+ in the interstitium (assumed proportional to *Ca* i.e = <sup>0</sup> *<sup>a</sup> k C* ) (in mOsm/L)

*kd* the transport coefficient of Na+ ions into the descending limb (in ml/min.mm)

*ka* the active transport coefficient of Na+ out of the ascending limb due to Na+ pump

*Q* is the tubular flow rate, assumed to be fairly constant in the first approximation (in mL/min)

The governing equations for the descending and ascending limbs of the loop of Henle are shown as a coupled system of linear first-order ODEs, with *Cd* and *Ca* the concentration/osmolality of the descending and ascending limbs of the loop of Henle respectively. In the descending limb, the change of concentration of Na+ is modelled as proportional to the concentration difference between the interstitium and the descending limb. In the ascending limb, the change of concentration of Na+ is modelled as directly proportional to the concentration in the ascending limb itself through active removal of Na+ by the Na+ pump. This leads to the following linear coupled system:

$$\text{Na} \leftarrow \text{in the descending limit:} \begin{aligned} \frac{d(\text{QC}\_d)}{d\mathbf{x}} &= k\_d \left(\text{C}\_i \cdot \text{C}\_d\right) = k\_d k\_0 \text{C}\_a \cdot \text{k}\_d \mathbf{C}\_d\\ \frac{d\mathbf{C}\_d}{d\mathbf{x}} &= \frac{k\_d}{Q} \left(\text{C}\_i \cdot \text{C}\_d\right) = \frac{k\_d k\_0}{Q} \text{C}\_a \cdot \frac{k\_d}{Q} \mathbf{C}\_d \end{aligned} \tag{1}$$

Na+ in the ascending limb: *<sup>a</sup> a a*

$$Q\frac{d\mathbf{C}\_a}{d\mathbf{x}} = k\_a \mathbf{C}\_a\tag{2}$$

with *k k d a* , 0 > . The flow rate Q in the renal tubule is taken as constant in the first approximation. Expressed as matrix equation with upper triangular matrix,

$$
\begin{pmatrix} \mathbf{C}\_{d} \\ \mathbf{C}\_{a} \end{pmatrix} = \begin{pmatrix} -\frac{k\_{d}}{Q} & \frac{k\_{d}k\_{0}}{Q} \\ 0 & \frac{k\_{a}}{Q} \end{pmatrix} \begin{pmatrix} \mathbf{C}\_{d} \\ \mathbf{C}\_{a} \end{pmatrix} \tag{3}
$$

The eigenvalues are <sup>−</sup> *dk Q* and *<sup>a</sup> <sup>k</sup> <sup>Q</sup>* and the eigenvectors are 1 0 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ and <sup>0</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>+</sup> <sup>⎣</sup> <sup>⎦</sup> *d d a k k k k* . The general solution of this system is given by:

$$
\begin{pmatrix} \mathbf{C}\_d \\ \mathbf{C}\_a \end{pmatrix} = H\_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} e^{-\frac{k\_d}{Q} \mathbf{x}} + H\_2 \begin{bmatrix} k\_d k\_0 \\ k\_d + k\_a \end{bmatrix} e^{\frac{k\_a}{Q} \mathbf{x}} \tag{4}
$$

where *H*1 and *H*2 are constants of the solution.

Analytically in phase space, since the 2 eigenvalues are real and opposite in sign, the origin of the linear system is a saddle point, asymptotically unstable. Hence, it is unlikely that the system will remain in the state of zero concentration in the ascending and descending limbs. In fact, the solution (4) shows that the system will tend towards a state where an increasing concentration exists in the loop of Henle, because of the positive eigenvalue *k Q <sup>a</sup>* / (representing active sodium transport in the ascending limb) for large *x*. This is consistent with the observation that it is the active sodium transport in the ascending limb that drives the production of the concentration gradient within the interstitium of the renal medulla and keeps the countercurrent mechanism operational, rather than the passive osmotic gradient as governed by −*k Q <sup>d</sup>* / which tends to dissipate the osmotic gradient.

At the loop end of the loop of Henle, the concentration/osmolality can reach extremely high levels, driven by active sodium transport. Indeed if the active Na+ transport = 0 *<sup>a</sup> k* , then the

*Q* is the tubular flow rate, assumed to be fairly constant in the first approximation (in

The governing equations for the descending and ascending limbs of the loop of Henle are shown as a coupled system of linear first-order ODEs, with *Cd* and *Ca* the concentration/osmolality of the descending and ascending limbs of the loop of Henle respectively. In the descending limb, the change of concentration of Na+ is modelled as proportional to the concentration difference between the interstitium and the descending limb. In the ascending limb, the change of concentration of Na+ is modelled as directly proportional to the concentration in the ascending limb itself through active removal of Na+

( ) ( )

( )

*d d d0 d*

*dC k kk k = C -C = C - C dx Q Q Q*

with *k k d a* , 0 > . The flow rate Q in the renal tubule is taken as constant in the first

⎛ ⎞

*k kk - C C Q Q <sup>=</sup> <sup>C</sup> k C <sup>0</sup>*

*d d0 d d a a a*

⎝ ⎠

*Q*

⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟

*<sup>Q</sup>* and the eigenvectors are

*d a k k x x d d Q Q*

*He H e*

1 2

*C k k* − ⎛ ⎞ ⎡⎤ ⎡ ⎤ ⎜ ⎟ = + ⎢⎥ ⎢ ⎥ <sup>+</sup> ⎝ ⎠ ⎣⎦ ⎣ ⎦

*a d a C k k*

Analytically in phase space, since the 2 eigenvalues are real and opposite in sign, the origin of the linear system is a saddle point, asymptotically unstable. Hence, it is unlikely that the system will remain in the state of zero concentration in the ascending and descending limbs. In fact, the solution (4) shows that the system will tend towards a state where an increasing concentration exists in the loop of Henle, because of the positive eigenvalue *k Q <sup>a</sup>* / (representing active sodium transport in the ascending limb) for large *x*. This is consistent with the observation that it is the active sodium transport in the ascending limb that drives the production of the concentration gradient within the interstitium of the renal medulla and keeps the countercurrent mechanism operational, rather than the passive osmotic

At the loop end of the loop of Henle, the concentration/osmolality can reach extremely high levels, driven by active sodium transport. Indeed if the active Na+ transport = 0 *<sup>a</sup> k* , then the

1 0

gradient as governed by −*k Q <sup>d</sup>* / which tends to dissipate the osmotic gradient.

*d QC =k C -C =k k C -k C*

*d i d d0 a d d*

(1)

*id a d*

*dC Q =kC dx* (2)

1 0 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

0

(3)

*k k* . The general

(4)

and <sup>0</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>+</sup> <sup>⎣</sup> <sup>⎦</sup> *d d a k k*

by the Na+ pump. This leads to the following linear coupled system:

*dx*

Na+ in the ascending limb: *<sup>a</sup> a a*

*Q*

where *H*1 and *H*2 are constants of the solution.

and *<sup>a</sup> <sup>k</sup>*

*d*

approximation. Expressed as matrix equation with upper triangular matrix,

' '

mL/min)

Na+ in the descending limb:

The eigenvalues are <sup>−</sup> *dk*

solution of this system is given by:

system decays to a baseline value through the exponential term associated with *dk* . This shows that without active transport, the concentration gradient and the countercurrent multiplier mechanism will dissipate within the renal medulla.

If we take the boundary conditions provided by empirical data in figure 2:

$$C\_d \left(0\right) = 300 \text{ mOSsm/L}$$

$$\mathbf{C}\_a(0) = 100 \text{ mOSsm/L}$$

and assuming trial values of *d a* = = 1 /mm *k k Q Q* , the concentration within the ascending and descending limbs are obtained as:

$$\begin{aligned} \mathbf{C}\_d &= 250e^{-\chi} + 50e^{\chi} \\ \mathbf{C}\_a &= 100e^{\chi} \end{aligned}$$

This is plotted in the following figures:

Fig. 5. Variation of Fluid Osmolality in the Descending and Ascending limbs of the Loop of Henle.

Although the values of kinetic transport coefficients are trial values, we can see how the model predicts the shape of the osmolality profile within the descending and ascending loops. The graphical representations in figure 5 demonstrate the highest level of osmolality achieved at the bottom end of the loop is about 1500 mOsm/L, reasonably consistent with empirical data. This model provides analytical solutions beyond that of Keener and Sneyd [3].

The limitations of this model is in the assumption of the values of transport coefficients of the descending and ascending tubules in the appropriate units, its linearity assumption and the disregard of its interaction with other modes or mechanisms of osmotic concentration. However, it can be seen that the analytical solutions provide a reasonable qualitative profile of the urinary concentration within the loop of Henle, under the assumptions made of its properties of its different segments and its "iterative" or "multiplier optimal design".

### **2.2 Single compartmental model of renal clearance kinetics – (1) single input**

One of the important function of the kidney is excretion of metabolic waste products. How the kidney handles this excretory function has direct implications on clinical or physiological function. It is thus of interest to analyse the behaviour of the kidney as an excretory system.

Most accessible to analysis is the renal response to a single bolus of a metabolic substrate. Most renal clearance kinetics analyse the behaviour of the excretory function of the kidney in respect to endogenous or exogenous substrates. In some assessments, it involves the administration of a single bolus dose of an exogeneous substance into the blood circulation.

When a single bolus of such a substance is introduced into the human body system though an intravenous injection, the substance will initially spread out in the circulatory system and distribute into the extravascular body-fluid compartments of the body, while it is at the same time being removed by the kidney. Hence, if we represent the human body as a two-compartment system, then there will be 2 phases of decrease of the plasma concentration of this substance. The first phase represents the fall due to rapid distribution of the substance within the body from the blood circulation into the equilibrium body fluid compartments of the body, while it is at the same time being removed by the kidney. The second phase represents the fall due largely to the renal excretion of this substance.

However, in most cases, the first phase can be ignored and corrected for by empirical approximation so that only the second slower phase needs to be measured. Hence, a single late exponential function can be used to describe the fall in the plasma concentration of the substance. This principle is used in the physiological measurement of renal clearance or glomerular filtration rate (GFR) in human subjects.

### **2.2.1 Renal clearance analysis using a single-bolus model of renal tracer or substrate**

Assume the amount of the tracer in the entire compartment is *A* (in mg or mmols). Let the concentration of the tracer in the compartment at time *t* be *Ct* (in mg/L or mmol/L) and the clearance be annotated as *g* (in L/min). By definition, *C = A/V,* where *V* is the total plasma volume or distribution volume, reasonably assumed constant in the body. By the principle of mass conservation,

Although the values of kinetic transport coefficients are trial values, we can see how the model predicts the shape of the osmolality profile within the descending and ascending loops. The graphical representations in figure 5 demonstrate the highest level of osmolality achieved at the bottom end of the loop is about 1500 mOsm/L, reasonably consistent with empirical data.

The limitations of this model is in the assumption of the values of transport coefficients of the descending and ascending tubules in the appropriate units, its linearity assumption and the disregard of its interaction with other modes or mechanisms of osmotic concentration. However, it can be seen that the analytical solutions provide a reasonable qualitative profile of the urinary concentration within the loop of Henle, under the assumptions made of its properties of its different segments and its "iterative" or

One of the important function of the kidney is excretion of metabolic waste products. How the kidney handles this excretory function has direct implications on clinical or physiological function. It is thus of interest to analyse the behaviour of the kidney as an

Most accessible to analysis is the renal response to a single bolus of a metabolic substrate. Most renal clearance kinetics analyse the behaviour of the excretory function of the kidney in respect to endogenous or exogenous substrates. In some assessments, it involves the administration of a single bolus dose of an exogeneous substance into the blood circulation. When a single bolus of such a substance is introduced into the human body system though an intravenous injection, the substance will initially spread out in the circulatory system and distribute into the extravascular body-fluid compartments of the body, while it is at the same time being removed by the kidney. Hence, if we represent the human body as a two-compartment system, then there will be 2 phases of decrease of the plasma concentration of this substance. The first phase represents the fall due to rapid distribution of the substance within the body from the blood circulation into the equilibrium body fluid compartments of the body, while it is at the same time being removed by the kidney. The second phase represents the fall due largely to the renal

However, in most cases, the first phase can be ignored and corrected for by empirical approximation so that only the second slower phase needs to be measured. Hence, a single late exponential function can be used to describe the fall in the plasma concentration of the substance. This principle is used in the physiological measurement of renal clearance or

**2.2.1 Renal clearance analysis using a single-bolus model of renal tracer or substrate**  Assume the amount of the tracer in the entire compartment is *A* (in mg or mmols). Let the concentration of the tracer in the compartment at time *t* be *Ct* (in mg/L or mmol/L) and the clearance be annotated as *g* (in L/min). By definition, *C = A/V,* where *V* is the total plasma

volume or distribution volume, reasonably assumed constant in the body.

This model provides analytical solutions beyond that of Keener and Sneyd [3].

**2.2 Single compartmental model of renal clearance kinetics – (1) single input** 

"multiplier optimal design".

excretion of this substance.

glomerular filtration rate (GFR) in human subjects.

By the principle of mass conservation,

excretory system.

$$\frac{dA}{dt} = -\mathbf{g} \cdot \mathbf{C}\_t \tag{5}$$

This is the governing first-order linear differential equation representing the kinetics of a one-compartment system.

Integrating over all time, the total tracer dose injected (*D*) is given by:

$$\int\_0^\infty \frac{dA}{dt} dt = \mathbf{D} = -\int\_0^\infty \mathbf{g} \mathbf{C}\_t dt = -\mathbf{g} \int \mathbf{C}\_t dt \tag{6}$$

The absolute magnitude of the renal clearance *g* is:

$$\left| \mathbf{g} \right| = \left| \frac{D}{\stackrel{\text{'''}}{\underset{\text{---}}{\rightleftarrows}}} \right| = \frac{\text{total dose of tracer injected}}{\text{area under the tracer concentration-time curve}} \tag{7}$$

We can show that the concentration of tracer in this compartment follows an exponential variation, by rewriting equation (5) as:

$$V\frac{d\mathbf{C}\_t}{dt} = -\mathbf{g}\cdot\mathbf{C}\_t\tag{8}$$

Separating variables, we get:

$$\int\_{C\_0}^{C\_t} \frac{dC\_t}{C\_t} = -\frac{\mathcal{g}}{V} \int\_{t\_0}^t dt$$

We have a mono-exponential clearance scheme, as follows:

$$\mathbf{C}\_{t} = \mathbf{C}\_{0}e^{-\frac{\mathbf{g}}{V}(t-t\_{0})} \tag{9}$$

By taking logarithms of both sides, we get a linear relationship on the "semi-log" scale as:

$$
\ln \mathbf{C}\_t = \ln \mathbf{C}\_0 - \frac{\mathcal{G}}{V} (t - t\_0) \tag{10}
$$

Equation (10) is the basis of plotting the tracer concentration against time as a semi-log graph, so that (i) the absolute value of the gradient of the slope will be given by (renal clearance)*/V,* which is also called the clearance constant *λ*, and (ii) the y-intercept will be given by *C0* which is *D/V*.

So the initial volume of distribution, *V*, will be given by

$$V = \frac{D}{C\_0} \tag{11}$$

Hence,

$$\text{Ranal clearance} = V \times \mathcal{A} = \frac{D}{C\_0} \times \mathcal{A} \tag{12}$$

Or, the estimated renal clearance is the

( ) ( ) total dose injected Distribution volume Clearance constant = gradient of ln C vs t curve y-intercept of ln C vs t curve × ×

Historically, this methodology is often known as the indicator-dilution method or the Stewart-Hamilton method, although the origins of this method antedate the work of Stewart and Hamilton [4,5].
