**3. Conclusion**

828 Biomedical Science, Engineering and Technology

demonstrate a higher blood concentration of creatinine for the same degree of renal

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

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

The result takes a useful form for physical interpretation. Total amount of substrate in the

( )

<sup>1</sup> <sup>−</sup> − − <sup>−</sup> ⎛ ⎞ ∗= ⋅ = − ⎜ ⎟

*g*

*gg g <sup>t</sup> <sup>t</sup> t u <sup>t</sup> VV V AV A e Adu e <sup>e</sup>*

0

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

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

∫

∗

*g t*

0

⎜ ⎟ ⎝ ⎠

∗= ⋅ ∫

− − −

*g g <sup>t</sup> <sup>t</sup> t u*

*At e <sup>V</sup>* (19)

( )

(21)

*A e Adu e V V* (20)

clearance.

have as follows:

given by:

body at time t is given by:

convolution analysis.

**2.3 Renal clearance – convolution analysis** 

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

Total amount of substrate in the body at time *t*: ( ) <sup>−</sup>

The analytical model of the loop of Henle and the renal handling of metabolic substrates is aimed at showing to some extent how the renal system is optimized for filtration and regulation of urine concentration by the countercurrent mechanism in the loop of Henle and its medullary environment, which is largely physiologically engineered to increase and maintain at steady-state the high osmolality of the urine fluid to as high as 4 times normal blood osmolality.

The renal clearance of substrates is modelled as a single-compartment kinetic model, with single-input and more physiologically as continuous-input. The analytical solutions obtained from the continuous input of creatinine predict the body creatinine level to tend to asymptotically steady-state substrate blood concentration with time, in a relationship that is an inverse rectangular hyperbolic function to the renal clearance. This is close to the relationship found empirically. The relationship is found to be related to the amount substrate input per unit time divided by renal clearance. The same conclusion is obtained from convolution analysis of renal clearance. The formula predicts reasonable estimates for the actual serum creatinine levels in the body based on renal clearance and substrate input parameters.
