**2. Materials and methods**

Literatures were searched for information on acute and chronic effects of lead bullets in wildlife, young and adult humans, especially in relation to brain damage and criminology. Some established formulas for determination of LD50 were modified for determination of acute and chronic toxicity effects of lead in wildlife and human. The developed formulas were used to determine lethal time fifty (LT50), chronicity factor (CF), dose of lead that can kill one man, toxic dose rate, target concentration, clearance, effective dose of antidote, volume of distribution, area under curve, mean residence time and elimination rate constant [21–23]. Up and down procedure (UPD) was used to estimate LD50 of bullets, as the initial estimate of the LD50 was within a factor of the authentic LD50 [24], and has been validated [25].

#### **2.1 Kinetic energy of lead bullet**

The velocity of bullet is proportionally more influential than the weight of the bullet. Therefore, the kinetic energy transferred to the target animal is presented by the equation given below [26]. Kinetic energy of bullet is the mass of bullet times squared velocity of the bullet.

$$\text{KE} = \frac{1}{2}\text{mv}^2\tag{1}$$

KE = kinetic energy; m = mass of bullet; v = velocity.

Low velocity = 1200 ft./s; medium = 1200–2500 ft./s; high = >2500 ft./s [27]. Bullets from handguns have velocity of <1000 ft./s, rifles (<2500 ft./s) and bullets of 5.56 mm are small but relatively fast. Outer jacket of the bullet determines margin line of tissue injury which is responsible for bullet moving more than 2000 ft./s [28].

Air rifle pellet that weighed 8.25 grains and of 0.38 calibre received velocity of 101 m/s, but could damage the brain, whereas that of 113 grains of 58 m/s could damage brain matter [29]. An impactor (bullet) of 200–297 mm<sup>2</sup> could exert force of 980–1334 N on parietal bone [30]. Therefore Head Injury Criterion (HIC) is used as protective measure for skull. It is a function of the period of acceleration at the head centre of gravity, bearing in mind that head is a one-mass structure:

$$\text{HIC} = \left\{ \left( t^2 - t^1 \right) \left[ \frac{\mathbf{1}}{t^2 - t^1} \int\_{t^2}^{t^1} a(t) \mathbf{d}t \right]^{2.5} \right\} \max \tag{2}$$

*t*<sup>1</sup> = initial time (s); *t*<sup>2</sup> = final time (s); *a*(*t*) = acceleration at the centre of gravity of the head; *t*2-*t*<sup>1</sup> = acceleration window (15–36 ms) [31].

#### **2.2 Determination of calcium-lead concentration in erythrocytes**

Erythrocyte volume ð Þ¼ *<sup>V</sup>* <sup>4</sup> <sup>3</sup> *πa*<sup>2</sup>*xb*, where *a* = larger axis; *b* = minor axis; *π* = 3.1415. Surface area and volume of a single red blood cell are 155 μm<sup>2</sup> and 87 μm<sup>3</sup> respectively [32]. Calcium concentration (mg/100 ml) to per cent decrease of lead erythrocytes content is equal to;

$$\frac{\text{@ decrease erythroçtes lead content}}{\text{concof Ca}\_2 \text{ mixture}} = \frac{40 - 10}{8 - 4} = 1:7.5\tag{3}$$

#### **2.3 Determination of acute toxicity of lead in rodents**

The detection limit of lead (0.04 μg/dl) at 5% level of significance, blood level lead (0.2–37 μg/dl) and urine lead (9–930 μg/dl) as well as less than 1% of lead transport from sink to plasma have been considered for calculations [21–23, 33, 34].

$$\text{Chronicity factor } (\text{CF}) = \frac{\text{LD}\_{50}}{\text{9Odays} \text{LD}\_{50}} \tag{4}$$

Dose rate DR ð Þ¼ Target concentration TC ð Þ� Clearance Cl ð Þ (5)

$$\text{Also median thermal time diffusivity} (\text{LT50}) = \frac{\text{LD}\_{50}}{D^{\text{P}}} \text{ whereas } D = \text{LD}; \tag{6}$$

*P* ¼ Power coefficient 1*=*3

$$\text{Note that ED50} = \frac{\text{LD}\_{50}}{3} \text{x} \text{W}\_a \text{x} \text{10}^{-4} \tag{7}$$

$$\frac{\text{LD}\_{50}}{\text{\\$}} = \frac{\text{ED}\_{50}}{W\_a \times 10^{-4}} \tag{8}$$

$$\text{LD}\_{50} = \frac{\text{3(ED\_{50})}}{W\_a \times 10^{-4}} \tag{9}$$

$$\text{LD}\_{50} = \text{LT}\_{50} \times D^P = \frac{\text{3(ED\_{50})}}{W\_a \times 10^{-4}} \tag{10}$$

$$D^P = \frac{\Im(\text{ED}\_{50})}{W\_d \times 10^{-4}} \times \frac{1}{\text{LT}\_{50}} \tag{11}$$

Therefore:

$$\text{ED}\_{50} = \frac{D^p \times \mathcal{W}\_a \times \mathbf{10}^{-4} \times \text{LT}\_{50}}{3} \tag{12}$$

$$\text{Also the volume of distribution } (\text{Vd}) = \frac{\text{Dose}}{\text{Concentration}} \tag{13}$$

$$\text{Clearance} = \frac{\text{Dose}}{\text{AUC}} \tag{14}$$

#### **2.4 Calculation of pulmonary oxygen toxicity of lead**

The cumulative pulmonary toxic dose (CPTD) of lead expressed as OTU (oxygen toxicity units) is calculated thus, partial pressure of oxygen (PO2) remains constant [35].

$$\text{OTU} = t\_{\text{x}} \mathfrak{x} \left( \frac{\text{0.5}}{\text{PO}\_{2} - \text{0.5}} \right)^{\frac{-5}{6}}. \tag{15}$$

Where as *tx* = time of exposure; PO2 = constant; �<sup>5</sup> <sup>6</sup> = exponent (�0.8333).

$$\text{Exposure time } t\_{\text{x}} = \text{Segmenttime } \mathbf{x} \left( \frac{\text{MaxPO}\_2 - \text{lowPO}\_2}{\text{MaxPO}\_2 - \text{MinPO}\_2} \right) \tag{16}$$

### **2.5 Calculation of central nervous system oxygen toxicity of lead**

CNS oxygen toxicity is calculated thus, PO2 remains constants [36]. Therefore

$$\text{CNSfraction} = \frac{\text{TimeatPO}\_2}{\text{Timelimit forPO}\_2} = \frac{\text{TimeatPO}\_2}{m \text{PO}\_2 + b} \tag{17}$$

where as *m* = slope of the time; *b* = intercept for the given range of PO2.

#### **2.6 Calculation of exposure dose of lead from contaminated environment**

The equation for calculation of exposure dose of lead from contaminated environment [35, 36] is given as follow:

$$D = \mathbf{C} \times \mathbf{IR} \times \mathbf{AF} \times \frac{\mathbf{EF}}{\mathbf{BW}},\tag{18}$$

whereas *D* = exposure dose; *C* = contaminated concentration; IR = intake rate of contaminated medium; AF = bioavailability factor; EF = exposure factor; BW = body weight, but the exposure factor is calculated as follow:

$$\text{EF} = \frac{(F \times \text{ED})}{\text{AT}},\tag{19}$$

whereas *F* = frequency of exposure (days/year); ED = Exposure duration (years); AT = averaging time (ED � 365 days/year).

### **2.7 Calculation of water dermal contact dose of lead**

Dose of water concentration of lead that can penetrate dermal layer of skin [35, 36] is calculated as follow:

$$D = \frac{(\text{C} \times P \times \text{BSA} \times \text{ET} \times \text{CF})}{\text{BW}},\tag{20}$$

whereas *D* = dose (mg/kg); *C* = contaminant concentration (mg/l); *P* = permeability coefficient (cm/h); BSA = exposed body surface area (m<sup>2</sup> ); ET = exposure time; CF = conversion factor (1 l/1000 cm3 ); BW = body weight.

#### **2.8 Calculation of soil ingestion exposure time of lead**

Dose of lead ingested via soil can be calculated [35, 36] using the following formula:

$$D = \frac{(\mathbf{C} \times \mathbf{IR} \times \mathbf{ET} \times \mathbf{CF})}{\mathbf{BW}},\tag{21}$$

where *D* = exposure dose (mg/kg); *C* = contaminant concentration (mg/kg); IR = intake rate of contaminated soil (mg/day); EF = exposure factor; CF = conversion factor (kg/mg); BW = body weight.

#### **2.9 Calculation of dose of lead particles present in food**

Quantity of lead fragments present in the ingested food can be calculated as follow:

$$D = \sum\_{n=i}^{n} \frac{\mathbf{C} \times \mathbf{Cri} \times \mathbf{EF}}{\mathbf{BW}},\tag{22}$$

whereas *D* = exposure dose (mg/kg/day); *C* = contaminant concentration (mg/g); Cri = consumption rate of incriminating food (g/day); EF = exposure factor; BW = body weight (kg); *n* = total number of incriminating food group [30].

#### **2.10 Toxicokinetic scaling of lead fragments**

Steady state volume of distribution (Vss) of lead fragments is calculated thus:

$$\text{Vss} = 1.22W^{0.68} \text{ where } W = \text{body weight of animal} \tag{23}$$

$$\text{Cl} = 0.91 \\ \text{W}^{0.5} \text{ where } \text{Cl} = \text{clearance} \\ \tag{24}$$

Vss and Cl are based on plasma concentrations of blood and free lead. However, the Vss and Clu for plasma free concentration of lead are presented below.

$$\text{Vss} = 247W^{0.93} \tag{25}$$

$$\text{Clu} = \mathbf{186} \mathcal{W}^{0.76} \tag{26}$$

Maximum tolerated dose (MTD) = 47*:*5*e*�0*:*51, *e* = 2.718 [37].

Relationship between respiratory minute volume and body weight is given by the equation.

$$V\_m = 0.518 W^{0.802} \tag{27}$$

A value of 15.6 l/min has been calculated for 70-kg weighed human [38].

Apparent volume of distribution (Vd) related to absolute oral bioavailability is given as follow:

$$\frac{\text{Vd}}{F} = \frac{\text{Dose}}{\text{AUC} \times \text{Kel}},\tag{28}$$

where *F* = bioavailability; AUC = area under curve; Kel = elimination constant.

$$\text{Cl}/F \times \text{MLP} = \beta \times W^{a} \tag{29}$$

where Cl = clearance; MLP = maximum lifespan potential; *W* = body weight; *a* = exponent.

$$\text{Dose} \left( \text{mg} \right) = \text{Animal AUC} \times \text{Scaled Human } \frac{\text{Cl}}{F} \tag{30}$$

AUC = lowest value among species [39].

Blood*=*Plasma concentration ratio Pp ð Þ¼ 1 þ H � ð Þ fu–1 (31)

whereas fu = fraction unbound in plasma; H = hematocrit (human, 0.44; rat,0.46; mouse, 0.45; rabbit, 0.36; monkey, 0.36).

$$\text{Cl} = \text{33.35ml/min} \times \frac{(a)^{0.77}}{\text{Rfu}} \tag{32}$$

Rfu = ratio of unbound fraction in plasma between rats and humans; *a* = coefficient of surface area.

$$\text{Conc.}H = \text{Conc.}Ax \frac{\text{Dose}\_H}{\text{Dose}\_A} \text{x} \left(\frac{\mathcal{W}\_A}{\mathcal{W}\_H}\right)^c \tag{33}$$

*Ecotoxicity Effects of Lead Bullets in Human and Wildlife: The Consequences… DOI: http://dx.doi.org/10.5772/intechopen.105850*

$$\text{Time}\_{H} = \text{Time}\_{A} \propto \left(\frac{W\_{A}}{W\_{H}}\right)^{b} \tag{34}$$

where *b* and *c* are exponents of simple allometry of Cl and Vdss, Percent (%) error between observed clinical concentration and predicted concentration of lead is calculated thus [40].

$$\% \text{error} = \frac{\text{Observed} - \text{Predicted}}{\text{Observed}} \times 100\tag{35}$$

The experiment for elimination half-life has allometric exponent of 0.25 [41].

Error involved in prediction of clearance is in most cases >30% [42]. Since inhaled lead particles can be distributed immediately the following equation can be used to calculate lead concentration in the blood.

$$\mathbf{C}\_{t} = \mathbf{C}\_{o}e^{-kt} \tag{36}$$

*Ct*= lead concentration at time *t*; *Co*= theoretical lead concentration obtained if it had been inhaled at time *t* = 0; *k* = elimination rate constant [43–45].

$$\text{Ratio of unitary clearance to total body clearance} = \frac{\text{Cl}\_{\text{u}}}{\text{Cl}\_{\text{b}}} \qquad \qquad (37)$$

Ratio of plasma free lead to total body lead <sup>¼</sup> *PL Bl* (38)
