**2. Materials and methods**

#### **2.1 Calculation of blood and plasma lead concentration**

Literatures were assessed with a view to obtaining formulas that could be used for the calculation of lead transport through brain capillaries and its pathogenesis of causing brain damage and low intelligence quotient. Measurement of red blood cell (RBC) partitioning of lead is done as follows:

$$\text{Plasma } \text{Imax}, u - \text{fun}\left( \text{Plasma } \text{Imax} + \frac{\text{F}\_a \times \text{F}\_\text{g} \times \text{K}\_a \times \text{Dose}}{\frac{\text{Qh}}{\text{B}b}} \right) \tag{1}$$

where *Imax* ¼ *C max* ð Þ *μmol* ,*Dose* ð Þ *μmol* ,*fup* ¼ fraction of unbound drug in plasma (lower limit, 0.01), *Ka*= absorption rate constant (0.1 mi*n*�<sup>1</sup>Þ, *Fa*= fraction absorbed (1), *Fg*= fraction escaping gut metabolism (1), *Qh*= hepatic blood flow (1.6 L/min), and *Rb*= blood–plasma concentration ratio [12].

$$\text{Distribution of lead to red blood cells } (\%) = \left(1 - \frac{Plasma\\_conc \times (100 - HC)}{Blood\\_conc \times 100} \right) \times 100 \,\text{L} \tag{2}$$

$$\text{Blood to plasma ratio} = \frac{\text{Blood Conc}}{\text{Plasma Conc}}\tag{3}$$

$$Hb = PV - \left(\frac{GrCl \times D \times Scr \times 72}{K \times (140 - Age)}\right) \times 12.5 \times \frac{1}{0.33} \tag{4}$$

$$D = \frac{Pcr}{\mathcal{Scr}} \times \mathbf{144.} \tag{5}$$

$$RDB = RRW \times 80 \text{ ml } 1 \text{kg} \times \frac{Desired \text{ } PCV - Recipient \text{ } PCV}{Donor \text{ } PCV} \tag{6}$$

#### **2.2 Blood and tissue kinetics of lead**

$$\mathsf{Cl} = \mathsf{DR} \langle \mathsf{C}s \rangle \tag{7}$$

$$LD = \text{Css} \times \text{Vd} \tag{8}$$

*Application of Lead Transport through Brain Capillary for Determination of Weight, Brain… DOI: http://dx.doi.org/10.5772/intechopen.107459*

The first-order equation that describes the release of lead from biological system is concentration-dependent and can be expressed as [13]:

$$
\log \mathcal{C}\_t = \frac{\log \mathcal{C}\_o - K\_t}{2.303} \tag{9}
$$

where *Co*= initial concentration of lead and *Ct*= concentration of lead in solution at time t. slope equals �*<sup>K</sup>=*<sup>2</sup>*:*<sup>303</sup> [14]. The elimination rate constant (1.16 h�<sup>1</sup> ) terminal half-life (0.6 h�<sup>1</sup> ) and volume of distribution (1.03 L/kg) have been reported for 60 kg weighted man.

Hence, terminal half-life

$$
\left(t\frac{1}{2}\beta\right) = \frac{Vd}{PCl} \times 0.693\tag{10}
$$

where *PCl* = plasma clearance. However, initial concentration (*Co*) is calculated as:

$$\mathcal{C}\_o = \frac{D}{V\_{app}}\tag{11}$$

where D = dose of lead; Vapp = apparent volume of distribution, when renal function is not impaired the serum concentration:

$$\mathbf{y}(\mathbf{C}\_{s}) = \frac{\text{observed concentration}}{\mathbf{0.2 \times Albumin (g/dl) + 0.1}} \tag{12}$$

When there is end-stage renal failure, the serum concentration is calculated as:

$$\text{C}\_{s} = \frac{observed\,\,concentration}{0.1 \times Albumin\,\,(g/dl) + 0.1} \tag{13}$$

Integration of exposure time with toxic dose of lead is presented as:

$$D = \frac{T^n}{K} \tag{14}$$

where D = dose, T = time of exposure, and K = concentration of toxicant causing toxicity [15].

#### **2.3 Translation of lead dose in animals to human**

$$HED = \frac{Animal\ dose \times Km}{Human\ Km} \tag{15}$$

Animal dose and animal Km are substituted with 40 and 20 kg as well as 31.3 and 25.0, respectively [16].

#### **2.4 Transport of lead through brain capillary**

Lead can be transported through brain capillary having the rate constant:

$$\mathbf{r}(\text{Kin}) = \frac{QE}{Vbrain} \tag{16}$$

$$E = \mathbf{1} - e^{\frac{-\text{PSA}}{Q}} \tag{17}$$

$$\text{Alternatively } E\_r = \frac{\text{Cin} - \text{Cout}}{\text{Cin}} \tag{18}$$

where Q = brain capillary blood flow, E = fraction of lead that flow into the brain, Vbrain = volume of brain, Er = extraction ratio of lead, PSA = permeability surface area ( *MS*�<sup>1</sup> *m*<sup>2</sup> of blood-brain barier, Cin = concentration of lead entering the brain capillary *mol L*�<sup>1</sup> , and Cout = concentration of lead leaving the brain capillary [17, 18].

The passive flux (QPas) of lead across blood-brain barrier between plasma and the brain extracellular fluid (ECF) is given as:

$$QPas = P\left(\mathbf{C}\_{pl} - \mathbf{C}\_{(ECF)}\right) \tag{19}$$

where QPas = passive flow rate *mol m*�<sup>2</sup>*s* �<sup>1</sup> ð Þ, P = permeability of the BBB *MS*�<sup>1</sup> , *Cpl*=concentration of the lead in the plasma *mol m*�<sup>3</sup> ð Þ, and *CECF*= concentration of lead in the ECF [19]:

$$\frac{\Delta \mathcal{C}\_{\rm{ECF}}}{\Delta t} = kBBB \left( \mathcal{C}\_{pl} - \mathcal{C}\_{\rm{ECF}} \right) \tag{20}$$

$$\text{V}\_{ECF} = \frac{\Delta(ECF)}{\Delta t} = \text{CLBBB} \left(\text{C}\_{pl} - \text{C}\_{ECF}\right) \tag{21}$$

$$\mathbf{C}\_{\rm{ECF}} = \frac{\mathbf{A}\_{\rm{ECF}}}{\mathbf{V}\_{\rm{ECF}}} \tag{22}$$

Cerebral metabolic rate (CMR) scales with brain volume, hence:

$$\frac{\text{CMR}}{V} = a \text{ V}^{0.167} \tag{23}$$

Density of neuron Dð Þ¼ *<sup>n</sup> <sup>α</sup> <sup>V</sup>*<sup>0</sup>*:*<sup>167</sup> (24)

$$\text{The capillary length density } (\mathbf{C}\_n) = a \, V^{0.167} \tag{25}$$

Total length of capillary is proportional to the number of neurons [20].

$$\text{Capillary diameter } (\mathbb{C}\_d) = a \; V^{0.08} \tag{26}$$

It is 7 *μm* diameter in human brain [21]. The flow rate of blood to the brain is 800 *mL= min* [22]where Δ*CECF* = change in lead concentration of extracellular fluid, KBBB = rate constant of lead transport across the BBB (*S*�<sup>1</sup> Þ, CL BBB = transfer clearance of lead transport across the BBB *m*<sup>3</sup>*S*�<sup>1</sup> , *AECF* = molar amount of lead in the brain ECF ð Þ *mol* , and *VECF*= volume of the brain ECF *<sup>m</sup>*<sup>3</sup> ð Þ [23–25].

$$\text{Passive permeability } (p) = P t \text{ means} + \frac{D\_{\text{para}}}{W\_{T\text{]}}} \tag{27}$$

*Application of Lead Transport through Brain Capillary for Determination of Weight, Brain… DOI: http://dx.doi.org/10.5772/intechopen.107459*

where Pt = passive transcellular permeability *ms*�<sup>1</sup> ð Þ, *Dpara* = diffusity of lead through the BBB intercellular space *m*2*s* �<sup>1</sup> ð Þ, and *WT*<sup>J</sup> = width of tight junction (m) [24].

$$\text{Total flux} \left( \mathbf{Q}\_{\text{total}} \right) = \mathbf{P}\_{\text{tot}} \left( \mathbf{C}\_{pl} - \mathbf{C}\_{ECF} \right) \tag{28}$$

$$\mathbf{Q}\_{\text{total}} = PA F\_{in} \left( \mathbf{C}\_{pl} \right) - PA F\_{out} \left( \mathbf{C}\_{ECF} \right) \tag{29}$$

where Ptotal = rate of active and passive transport across BBB, *PAFin* = affinity of lead to active transport into the brain, and *PAFout* = affinity of lead to active transport out of the brain [25].

$$P\_{\text{tot}} = P\_{\text{pass}} \times \mathcal{C}\_{\text{part}} \tag{30}$$

*Ppas* = passive permeability to BBB; *Cpart*= coefficient of partition [26].

$$\text{Active clearance } (Cl\_{act}) = \frac{Tm}{Km + C} \tag{31}$$

where *Tm* = maximum rate of lead transport across the BBB ( *μmol L*�<sup>1</sup> *S*�<sup>1</sup> , K*m* = concentration of free lead *μmol L*�<sup>1</sup> at which half of *Tm* is attained, and C = concentration of lead in plasma [27].

Change in lead concentration within the cells of the brain is given as follows:

$$\frac{\Delta \mathbf{C}\_{ICF}}{\Delta t} = \mathbf{K}\_{cell} (\mathbf{C}\_{ECF} - \mathbf{C}\_{ICF}) \tag{32}$$

$$V\_{ICF} \frac{\Delta C\_{ICF}}{\Delta t} = C\_{cell} (C\_{ECF} - C\_{ICF}) \tag{33}$$

$$\mathbf{C}\_{ICF} \frac{\mathbf{A}\_{ICF}}{\mathbf{V}\_{ICF}} \tag{34}$$

where *CICF* = concentration of lead in the brain ICF *μmol L*�<sup>1</sup> , *Kcell*= rate constant of lead transport across the cell membrane *S*�<sup>1</sup> , and *VICF*= apparent volume of distribution in the brain ICF [28].

$$\text{Dissociation constant } (kd) = \frac{k\_{on}}{k\_{off}} \tag{35}$$

where *kd* ¼ 45*:*62 *nM* ð Þ 0*:*91 *μg=dL* that has been reported for on-site lead detection using a biosensor device [29].

#### **2.5 Michaelis-Menten kinetics of lead**

Clearance of lead by Michaelis-Menten kinetics is given as follows:

$$\text{CLact} - cell = \frac{\text{Tm} - cell}{\infty (\text{Km} - cell + \text{C})} \tag{36}$$

where *CLact* � *cell* = active transfer clearance of free lead across the cell membrane, C = concentration of lead in the brain ECF or ICF, *Tm* � *cell*= maximal veolocity of the transporter, and *Km* � *cell*= Michaelis –Menten constant [30].

Enzyme metabolic clearance ð Þ *φmet* is given as follows:

$$
\rho \text{met} = V\_{\text{max}} \frac{\text{C}}{\text{K}m + \text{C}} \tag{37}
$$

where *φmet* = flux of the enzymatic metabolic reaction *mmol L*�<sup>1</sup> *min*�<sup>1</sup> , *V max* = maximum flux of the reaction *mmol L*�<sup>1</sup> , C = concentration of substrate in ECF or ICF ( *mmol L*�<sup>1</sup> , and *Km* = affinity of coefficient of the enzyme substrate *mmol L*�<sup>1</sup> [31]. Three-dimensional model that integrates lead transport through BBB and lead binding within the brain could predict lead distribution in the brain [32]. One kilogram equals 1000 mililiters [33].

#### **2.6 Relationship between brain mass and encephalization quotient**

$$\text{Enceptible quotient (EQ)} = \frac{\text{Brain mass}}{0.14 \times \text{Body weight}^{0.528}} \tag{38}$$

$$\text{Brain mass } (\text{E}) = k\_p \beta \tag{39}$$

where k = 0.14, p = body weight, and β = 0.528 [34, 35].

$$\text{Brain volume} \left( \log\_{10}(B) \right) = 3.015 + 0.986 \log\_{10} C \tag{40}$$

where B = brain size (mm3 ) and C = internal cranial capacity (mm<sup>3</sup> ).

$$\text{Also brain volume } (V\_{b\text{train}}) = \forall\_3 \times \pi \times r^3 \tag{41}$$

where

$$
\pi = 3.14159 \text{ and } \mathbf{r} = \text{radius} = \frac{diameter}{2} \tag{42}
$$

$$T\frac{1}{2}\beta = \frac{0.693}{\beta} \tag{43}$$

Lead concentration (Ct), plasma clearance (Pcl), calculated administered lead and time of exposure to lead were extrapolated to 60, 40, and 20 kg weighed human, using human equivalent dose formula.
