**2.2.2 Physiological measurement of the Glomerular Filtration Rate (GFR)**

If we can insert a microneedle with a flow gauge into each glomerulus in the kidney and measure the flow rate experimentally, this would constitute one way of measuring the GFR. This can be done in-vitro with micropuncture and microperfusion techniques. On a body system level, the global renal clearance has to be obtained by other ways.

Typically, GFR is deduced by measuring the renal filtered loss or clearance of a suitable substance from the plasma. In physiological tests, creatinine clearance and Cr-51 EDTA clearance are typically used. Dynamic renogram modelling of impulse tracer kinetics through the kidney had previously been performed [6].

 In order to measure GFR accurately, the following properties must apply to the particular substance used to measure GFR :


 These are severe restrictions and there are only some possible candidates for this substance, including :


If the loss of these substances from the body can be measured, one can get a quantitative reflection of the excretory function of the kidney.

Of the four substances mentioned, only endogenous creatinine is found within the human body. The other substances have to be introduced into the human body. Inulin is a polysaccharide molecule with a molecular weight of 5200. Creatinine itself is a by-product of skeletal muscle metabolism, and it is present in the plasma at a relatively constant concentration and does not require intravenous infusion into the patient. However, creatinine is not a perfect marker for GFR because a small amount of it is secreted by the tubules and hence it is not a pure glomerular agent and tends to overestimate the GFR.

Incidentally, there is an agent, para-aminohippuric acid (PAH), which is not only filtered but also secreted to a large extent, so that it can be used to measure not the GFR but the effective renal plasma flow rate. Chromium-51 EDTA is a radiolabelled EDTA with the gamma emitter, chromium-51. This agent is very close to a purely glomerular filtered agent. Tc99m-DTPA is also a glomerular filtered substance, radiolabelled to the gamma emitter, Tc99m.

### **2.2.3 Continuous input of substrate model – relationship between steady-state serum creatinine concentration in the body and renal clearance**

### **2.2.3.1 Theory and application**

824 Biomedical Science, Engineering and Technology

Renal clearance =×= ×

( ) ( ) 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

If we can insert a microneedle with a flow gauge into each glomerulus in the kidney and measure the flow rate experimentally, this would constitute one way of measuring the GFR. This can be done in-vitro with micropuncture and microperfusion techniques. On a body

Typically, GFR is deduced by measuring the renal filtered loss or clearance of a suitable substance from the plasma. In physiological tests, creatinine clearance and Cr-51 EDTA clearance are typically used. Dynamic renogram modelling of impulse tracer kinetics

In order to measure GFR accurately, the following properties must apply to the particular

These are severe restrictions and there are only some possible candidates for this substance,

If the loss of these substances from the body can be measured, one can get a quantitative

Of the four substances mentioned, only endogenous creatinine is found within the human body. The other substances have to be introduced into the human body. Inulin is a polysaccharide molecule with a molecular weight of 5200. Creatinine itself is a by-product of skeletal muscle metabolism, and it is present in the plasma at a relatively constant concentration and does not require intravenous infusion into the patient. However, creatinine is not a perfect marker for GFR because a small amount of it is secreted by the tubules and hence it is not a pure glomerular agent and tends to overestimate the GFR. Incidentally, there is an agent, para-aminohippuric acid (PAH), which is not only filtered but also secreted to a large extent, so that it can be used to measure not the GFR but the effective

**2.2.2 Physiological measurement of the Glomerular Filtration Rate (GFR)** 

system level, the global renal clearance has to be obtained by other ways.

through the kidney had previously been performed [6].

1. the substance must be freely filtered through the glomerulus 2. it must not undergo renal tubular secretion or absorption

4. it must not be lost through any other methods from the body 5. it must not be metabolised or changed chemically in the body.

1. endogenous creatinine, but this is less accurate in children

reflection of the excretory function of the kidney.

substance used to measure GFR :

including :

2. inulin

3. chromium-51 EDTA 4. technetium99m-DTPA

3. it must not bind to plasma proteins

0

 λ

*C* (12)

λ

*<sup>D</sup> <sup>V</sup>*

Hence,

and Hamilton [4,5].

Or, the estimated renal clearance is the

As opposed to the single-bolus renal kinetics, in the body, endogenous metabolic substrates are introduced into the blood circulation in a continuous way. Thus the model of renal clearance kinetics given above will have to be modified to take this continuous input into account. Analysing this continuous input model will be useful to evaluate the optimal and crucial renal handling of endogenous waste products in a typical human body.

The result of such a continuous input and renal excretion gives rise to a steady-state concentration of a renal-excreted substrate in the body. A typical endogenous substrate produced in a continuous fashion in the body and excreted by the renal route is creatinine. Figure 6 shows an inverse relationship between plasma creatinine concentration and GFR. The lower the renal clearance, the higher is the steady-state blood concentration of the substrate. However, this relationship is not linear but largely inverse rectangular hyperbolic (see figure 6).

To analyse this empirical relationship, further analysis can be performed using the singlecompartment model but introducing a continuous input of substrate. This analysis follows from and extends the results obtained by Mazumdar [7].

Fig. 6. Empirical relationship between blood creatinine levels and the renal clearance rate (adapted from [2]).

If instead of a single dose of tracer or substrate given as discussed in the previous section, constant doses of creatinine (substrate) are given at equal intervals of time ie, at intervals of period *T*, then using the single-bolus equation (7), the concentration of the substrate immediately after the second dose is given by:

$$\mathbf{C}\_{1} = \mathbf{C}\_{0} + \left(\mathbf{C}\_{0}e^{-\frac{\mathcal{S}}{V}T}\right) \tag{13}$$

Immediately after the third dose, the substrate concentration is given by:

$$\begin{aligned} \mathbf{C}\_2 &= \mathbf{C}\_0 + \left[ \mathbf{C}\_0 + \begin{pmatrix} \mathbf{C}\_0 e^{-\frac{\mathcal{S}}{V}T} \\ \mathbf{C}\_0 e^{-\frac{\mathcal{S}}{V}T} \end{pmatrix} \right] e^{-\frac{\mathcal{S}}{V}T} \\ &= \mathbf{C}\_0 + \mathbf{C}\_0 e^{-\frac{\mathcal{S}}{V}T} + \mathbf{C}\_0 e^{-\frac{2\mathcal{S}}{V}T} \end{aligned} \tag{14}$$

Immediately after the *n*th dose, the substrate concentration is given by:

$$\begin{aligned} \mathbf{C}\_{n-1} &= \mathbf{C}\_0 + \mathbf{C}\_0 e^{-\frac{\mathcal{S}}{V}T} + \dots + \mathbf{C}\_0 e^{-(n-1)\frac{\mathcal{S}}{V}T} \\ &= \mathbf{C}\_0 \left[ 1 + e^{-\frac{\mathcal{S}}{V}T} + \dots + e^{-(n-1)\frac{\mathcal{S}}{V}T} \right] \\ &= \mathbf{C}\_0 \frac{1 - e^{-n\frac{\mathcal{S}}{V}T}}{1 - e^{-\frac{\mathcal{S}}{V}T}} \end{aligned} \tag{15}$$

As *n* tends to infinity, the creatinine concentration approaches an equilibrium value, given by:

$$C\_{\infty} = \frac{C\_0}{1 - e^{-\frac{\mathcal{S}}{V}T}} \tag{16}$$

Linearization by Taylor's series approximation gives the equilibrium concentration of the creatinine in the blood to first order, as:

$$C\_{\infty} = \frac{C\_0}{\frac{\mathcal{S}}{V}T - \frac{1}{2}\left(\frac{\mathcal{S}}{V}T\right)^2 + \dots} = \frac{(C\_0 \,/\, T)V}{\mathcal{S}}\tag{17}$$

where ( ) *C TV* <sup>0</sup> / is the amount of the creatinine introduced per unit time. Hence, using the parameter values in Table 1,

$$C\_{\infty} = \frac{\text{amount of real substrate introduced per unit time}}{\text{real clearance}} = \frac{10.1}{\text{real clearance}}\tag{18}$$

If instead of a single dose of tracer or substrate given as discussed in the previous section, constant doses of creatinine (substrate) are given at equal intervals of time ie, at intervals of period *T*, then using the single-bolus equation (7), the concentration of the substrate

> <sup>−</sup> ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

> > 2

( )

− − −

...

1

*g g T nT*

− −

*g g T T V V*

( )

1

*g T*

*C C Ce <sup>V</sup>* (13)

(14)

(15)

(16)

(17)

10 0

20 0 0

00 0

100 0

= + ++ ⎡ ⎤ = + ++ ⎢ ⎥ ⎣ ⎦

*C C Ce Ce*

− −−

*V V <sup>n</sup> g g T nT V V*

As *n* tends to infinity, the creatinine concentration approaches an equilibrium value, given

1 <sup>∞</sup> <sup>−</sup> =

*<sup>C</sup> <sup>C</sup>*

−

Linearization by Taylor's series approximation gives the equilibrium concentration of the

2

*g g <sup>g</sup> T T*

where ( ) *C TV* <sup>0</sup> / is the amount of the creatinine introduced per unit time. Hence, using the

amount of renal substrate introduced per unit time 10.1

*C*<sup>∞</sup> ≈ = (18)

<sup>1</sup> ... <sup>2</sup>

<sup>∞</sup> = ≈ ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠

*V V*

*e*

0

*g T V*

<sup>0</sup> ( ) <sup>0</sup>

*C C TV*

/

renal clearance renal clearance

1 ...

*Ce e*

*<sup>g</sup> n T V g T V*

−

−

*e*

*C Ce Ce*

=+ +

Immediately after the *n*th dose, the substrate concentration is given by:

0

−

0

creatinine in the blood to first order, as:

parameter values in Table 1,

*C*

by:

<sup>−</sup> <sup>=</sup>

1

*<sup>e</sup> <sup>C</sup>*

1

−

− −

=+ + ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

*C C C Ce e*

⎡ ⎤ ⎛ ⎞

*g g T T V V*

Immediately after the third dose, the substrate concentration is given by:

immediately after the second dose is given by:

*C*∞ is the steady-state concentration of creatinine, as it is produced and excreted continuously in the body. The relationship derived is the equilibrium or steady-state concentration of the substrate that is produced continuously in the body and excreted renally. The relationship is plotted in figure 6.


Table 1. Important physiological renal parameters (data from [2]).

Fig. 7. Model prediction of the relationship between blood creatinine levels and the renal clearance rate.

Graphically, equation (17) predicts a very close approximation to the empirical curve, which is an inverse rectangular hyperbolic relationship between serum creatinine levels and the renal clearance as shown in figure 7.

Depending on the rate of production of the metabolite creatinine in the human body, equation (17) also demonstrates that there is a series of iso-dose curves of renal clearance vs blood levels of renal substrate, similar to isothermal curves in ideal gas thermodynamics.

The application to human physiology can be seen as follows. It is well known that the serum creatinine levels in women is lower than in man. A typical muscular man producing a larger quantity of creatinine substrate due to muscle breakdown will, for the same renal clearance, demonstrate a higher blood concentration of creatinine for the same degree of renal clearance.

### **2.3 Renal clearance – convolution analysis**

In general, the total amount of substrate in the body at time *t* is given by the convolution of the amount produced by the body per unit time, *A(t)*, which is a function of time and the biological clearance of that substance. In the case of pure renally-excreted substrate, such as creatinine, assuming a single-compartment clearance-kinetics as previously discussed, we have as follows:

Total amount of substrate in the body at time *t*: ( ) <sup>−</sup> ∗ *g t At e <sup>V</sup>* (19)

If as above, the amount of substrate introduced into the blood compartment per unit time is constant, *A*, then the total amount of substrate at time *t* (accounting for renal clearance) is given by:

Total amount of substrate in the body at time *t*:

$$A \ast e^{-\frac{\mathcal{S}}{V}t} = \int\_0^t A d\mu \cdot e^{-\frac{\mathcal{S}}{V}(t-\mu)}\tag{20}$$

The result takes a useful form for physical interpretation. Total amount of substrate in the body at time t is given by:

$$A \ast e^{-\frac{\mathcal{S}}{V}t} = \int\_0^t A du \cdot e^{-\frac{\mathcal{S}}{V}(t-u)} = \frac{AV}{\mathcal{S}} \left(1 - e^{-\frac{\mathcal{S}}{V}t}\right) \tag{21}$$

Schematically, this relation is shown in figure 8, for blood creatinine levels:

Fig. 8. Asymptotic steady-state concentration of blood creatinine levels, based on convolution analysis.

At *t* → ∞ , the equilibrium concentration of substrate in the blood compartment is:

$$C\_{\infty} = \frac{1}{V} \left(\frac{AV}{g}\right) = \frac{\text{amount of substrate introduced per unit time}}{\text{real clearance}}\tag{22}$$

consistent with the previous equation (18).

Application of the formula can be done using the physiological values in Table 1.

The body produces creatinine at the rate of 20-25 mg/kg body weight per day, which is approximately 1.5 g/day in a 70 kg man. In SI units, this is 10.1 umol/min. The renal clearance is approximately 120 ml/min. Hence, based on equation (22), the estimated steady-state serum creatinine level in the body based on the model would be predicted to be in the order of

10.1/120 = 0.084 umol/ml or 84 umol/L, as expected empirically.

Direct correlation with physiological parameters shows this convolution analysis to give reasonably close results.
