#### 3.2.2. Equivalent Dose of NP and Weighting Factors

In analogy to radiation weighting factors (w_{R}) for deposited radiation energy, we now propose weighting factors (w) which relate to specific quantifiable physico-chemical (PC) properties of the NP. Hence, and in contrast to the w_{R} in radiation dosimetry, there will be several weighting factors w_{PC,i} (i = 1, 2, ...) for NP which may be constants or functions of NP parameters.

Accordingly, the equivalent dose

H_{NP}_{,T} for each organ or tissue T will be defined similarly to Equation (2). In other words, we will use the term “equivalent dose for NP” considering the “bioactivity” (or a precursor of equivalent dose) of the NP. The equivalent dose is the product of

w_{PC,i} multiplied by the deposited dose. Since the weighting factors have been assigned to characteristics of the NP which all influence the total absorbed dose, here the product of weighting factors has been chosen to enable that each weighting factor separately influences the dose according to the biological impact of the respective NP characteristics.

Table 1 shows a comparison between the equations for ionizing radiation and NPs:

Introducing this concept of weighting factors w for different PC properties of NPs will allow for the estimation of the equivalent dose H_{NP,T} as a function of the deposited NP surface area and of equivalent dose comparisons between various engineered NPs and their PC properties. We consider that different NP properties are associated with the induction of biological effects. The majority of studies which have investigated biological endpoints in nanotoxicology find clear correlations between the induction of ROS and nitrogen radicals and our dose metric, namely the deposited NP surface area.

With this dose concept for deposited NPs using the total deposited surface area as the dose metric, a rationale-based and more quantitative ranking of possible effects of NPs in biological tissues will be possible. As PC properties characterized by weighting factors we especially consider: (1) the specific surface area, (2) the surface textures, (3) the zeta-potential as a measure for surface charge and (4) the particle morphology such as the shape and the length-to-width ratio (aspect ratio). (5) Furthermore, the band gap energy levels of metal and metal oxide NP are considered where oxidative stress is caused by NPs. The oxidative stress is based on the relationship between the cellular redox potential and the band gap energy levels. A further relevant physico-chemical property is (6) the particle dissolution/dissociation rate. The proposed w_{PC,i} can be extended and/or modified and adapted according to the available data and emerging knowledge. In the present model the reactivity of free radicals is considered to be the most relevant biological response for dose estimation. Furthermore, functionalized and/or surface modified NPs may represent “new” NPs with their own surface area and physico-chemical properties.

1. Proposal for the first weighting factor w_{PC}_{,1}: The specific surface area (SSA), i.e., the ratio of NP surface area to the NP volume, increases with decreasing particle diameter. One single nanoparticle can contain atoms/molecules from a few to several orders of magnitude higher numbers of atoms/molecules. As a result, a certain fraction of the atoms/molecules is located on the particle surface depending on the size, the shape and the physical structure like roughness/smoothness, etc. Since it has been shown in many recent papers that the specific surface area is most relevant for the induction of oxidative stress, we suggest the first weighting factor w_{PC,1} to be related to the fraction of atoms/molecules on the surface of NPs relative to the total number of atoms/molecules.

For example, for a spherical NP with the diameter d the volume is π/6 d

^{3}, and the spherical surface area SA is π d

^{2}; hence, the specific surface area SSA becomes:

Then the weighting factor is a function of the inverse NP diameter d:

In the following we show how the fraction of atoms/molecules on the NP surface

N_{SA} per total number of atoms/molecules

N_{tot} within the NP will change with NP size

d_{NP} and the atomic/molecular volumes

V_{mol} of a number of relevant metal and metal oxide NPs. This fraction will be proportional to the weighting factor affecting biological effects:

This ratio is shown for gold (atom diameter 0.29 nm) and silver (atom diameter 0.32 nm) in

Figure 1a. Since the gold atoms are smaller than the silver atoms, the fraction of gold atoms on the surface is slightly lower compared to that of the silver atoms.

Similarly, crystallographic data on the molecular dimensions can be used to estimate the number of various metal oxides molecules on the surface as well as within the NP lattice; this is shown for spherically shaped NPs of titanium dioxide, gamma Fe

_{2}O

_{3} hematite, Fe

_{3}O

_{4} magnetite and monoclinic CuO in

Figure 1b. Since the tetrahedral structure of the TiO

_{2} molecules is smallest, their fraction of atoms on the surface relative to the total number of TiO

_{2} molecules in the NP is smaller than those of the other metal oxide molecules; vice versa the surface fraction of Fe

_{3}O

_{4} magnetite is highest since these are the largest of the selected metal oxide molecules.

While this

w_{PC}_{,1} is declining rapidly with increasing size of the NP (see

Figure 1a,b),

w_{PC}_{,1} behaves different for

agglomerates of primary nano-scaled particles. Agglomerates coagulate, e.g. due to weak forces like the van-der-Waals force or magnetic forces in case of magnetic nanoparticles, such that their contact points are infinitesimally small [

30]. As a result the surface area

SA_{agg} of an agglomerate is the sum of the surface areas of all primary particles

SA_{pp} within an agglomerated particle; similarly the volume of the aggregate is the sum of all volumes of the primary particles. If all primary particles are of the same size then they all have the same surface area

SA_{pp} and the same volume

V_{pp} or mass and, hence, the same specific surface area

SSA_{pp}. Applying this to an agglomerated particle of n identical primary particles, the agglomerated particle has the same specific surface area

SSA_{pp} as the primary particles independent of the number of primary particles within an agglomerated particle:

Hence

w_{PC}_{,1} becomes a constant:

where

c_{agg} is a constant which differs for different sizes of primary particles.

If the primary particles are not all of the same size but show a size distribution, then their agglomerates may differ in their specific surface area. Yet when considering a large enough number of agglomerates, they will have a specific surface area

SSA_{agg} which is equal to the average of the specific surface area

SSA_{pp} of the primary particles and this is independent of the actual size of the agglomerate [

31].

**Figure 1.**
Weighting factor 1 in arbitrary units which is proportional to SSA—i.e., the ratio of atoms/molecules at the NP surface over the total number of atoms/molecules: (**a**) gold (Au) and silver (Ag) NP, (**b**) TiO_{2}, CuO, Fe_{2}O_{3}, and Fe_{3}O_{4}, (**c**) for agglomerated NPs with 2, 5 and 10 nm primary particle size, (**d**) spherical and agglomerated TiO_{2} with different primary sizes. Panels (c) and (d) show that w_{PC,1} is constant for different sizes of agglomerates but changes with the SSA of primary particles.

**Figure 1.**
Weighting factor 1 in arbitrary units which is proportional to SSA—i.e., the ratio of atoms/molecules at the NP surface over the total number of atoms/molecules: (**a**) gold (Au) and silver (Ag) NP, (**b**) TiO_{2}, CuO, Fe_{2}O_{3}, and Fe_{3}O_{4}, (**c**) for agglomerated NPs with 2, 5 and 10 nm primary particle size, (**d**) spherical and agglomerated TiO_{2} with different primary sizes. Panels (c) and (d) show that w_{PC,1} is constant for different sizes of agglomerates but changes with the SSA of primary particles.

Hence, for agglomerates containing a large enough number of primary particles with a mean

SSA_{pp} w_{PC}_{,1} is also a constant

c’_{agg} depending solely on the mean

SSA_{pp} of the primary particles (

Figure 1c,d)

2. Proposal for the second weighting factor w_{PC}_{,2}: w_{PC}_{,2} represents an arbitrary weighting factor for the surface texture. For spherical NPs with smooth surfaces w_{PC}_{,2} is set to unity w_{PC}_{,2} = 1 while w_{PC}_{,2} becomes w_{PC}_{,2} > 1 for irregularly shaped NPs with sharp edges and ridges and a rough surface. This surface roughness relates to the fact that atoms/molecules on the rough surface of NP are less integrated into the NP lattice and may be “unsaturated”, such that they are more prone to electron transfer than the inner atoms/molecules of the NPs leading to oxidative stress reaction in biological systems. In addition, their relative number to the total number increases due to the increasing surface area; hence, free radical production may increase and/or these atoms/molecules may change their stoichiometry and/or they may even leave the NP surface such that the remaining NP surface texture may increase in surface reactivity.

For spherical NPs curvature increases drastically with decreasing size and is proportional to the inverse of the diameter

d of the NP as discussed above. In analogy of the Young-Laplace equation which relates pressure

p on the surface of a liquid sphere to surface tension σ and the inverse of the diameter:

we can introduce a surface module σ’ which describes the probability of atoms/molecules at the surface to either alter their electron shell or change their molecular composition (which also leads to altered electron shells) or even escape the surface. The probability will be greater than the probability σ

_{0}‘ for atoms/molecules inside the NP. In the absence of precise data we propose for

w_{PC,2}:

This is shown in

Figure 2 for a NP with smooth surface

versus a NP with a 10-times increased roughness.

3. Proposal for the third weighting factor

w_{PC,3}: The

zeta potential (ZP) is the net electric potential formed by the charged groups of molecules of the NP surface and the surrounding medium. The ZP depends strongly on the pH of the medium: Commonly, absorption of proteins onto the NP surface may shift the ZP towards neutral ZP. For a quantitative estimate of ZP we use data recently published [

32,

33] and plot the relative neutrophil influx (NNI) in rat lungs

versus the ZP (mV) of NPs of different materials which had been instilled 24 hours prior to broncho-alveolar lavage and subsequent NNI and ZP measurements.

The neutrophil influx is normalized to baseline values of neutrophil influx observed with non-reactive NP and without NP. The observed increase of NNI was fitted by a linear trend [

32,

33] (

Figure 3):

Therefore, we propose for the zeta potential ZP a weighting factor w

_{PC,3} (arbitrary value):

**Figure 2.**
Weighting factor 2 (w_{PC,2}) is proportional to the surface texture of NPs given as relative roughness module: spherical NPs with smooth surface w_{PC,2} = 1 while w_{PC,2} is set to 10 for irregularly shaped NPs with sharp edges and ridges and a rough surface.

**Figure 2.**
Weighting factor 2 (w_{PC,2}) is proportional to the surface texture of NPs given as relative roughness module: spherical NPs with smooth surface w_{PC,2} = 1 while w_{PC,2} is set to 10 for irregularly shaped NPs with sharp edges and ridges and a rough surface.

**Figure 3.**
Weighting factor 3 (

w_{PC,3}) is depending on the zeta potential: The weighting factor is proportional to the relative PMN influx and increases according to equation 16. However, for a low zeta potential, it is not less than 1 [

29,

32].

**Figure 3.**
Weighting factor 3 (

w_{PC,3}) is depending on the zeta potential: The weighting factor is proportional to the relative PMN influx and increases according to equation 16. However, for a low zeta potential, it is not less than 1 [

29,

32].

4. Proposal for the fourth weighting factor

w_{PC,4}: The

w_{PC,4} is a weighting factor for the

particle morphology. NPs may not only be spherically shaped, but elongated with a very high aspect ratio (ratio of NP length to NP diameter) like biopersistent and long carbon nanotubes (CNTs). We gather from the existing literature on CNTs and biopersistent asbestos fibers that the induction of free radicals and hence oxidative stress is caused by frustrated phagocytosis of lung macrophages ([

33] and others) which cannot completely phagocytize CNTs with a length of more than 15 µm. This phenomenon has been shown in the literature and represents therefore a clear connection to the development of mesothelioma and subsequent death as shown by asbestos workers in the past. As an example we set normal phagocytosis without enhanced induction of ROS to unity and compare this to a high factor of 500 for enhanced frustrated phagocytosis associated with massive ROS induction and the potential for carcinogenicity. Due to the carcinogenic risk of developing mesothelioma we propose a high value for the weighting factor

w_{PC,4} (

Figure 4):

**Figure 4.**
Weighting factor 4 (w_{PC,4}) for particle morphology is proportional to frustrated phagocytosis of intratracheally instilled biopersistent and long carbon nanotubes (CNT): for normal phagocytosis w_{PC,4} is set to 1 while the w_{PC,4} is set to 500 for enhanced frustrated phagocytosis of CNT longer than 15 μm. We consider the risk of cancer induction and the risk of subsequent death as a most threatening event such that we chose a factor of 500.

**Figure 4.**
Weighting factor 4 (w_{PC,4}) for particle morphology is proportional to frustrated phagocytosis of intratracheally instilled biopersistent and long carbon nanotubes (CNT): for normal phagocytosis w_{PC,4} is set to 1 while the w_{PC,4} is set to 500 for enhanced frustrated phagocytosis of CNT longer than 15 μm. We consider the risk of cancer induction and the risk of subsequent death as a most threatening event such that we chose a factor of 500.

5. Proposal for the fifth weighting factor

w_{PC,5}: Burello and Worth [

9] developed a theoretical predictive model for oxidative stress caused by metal and metal oxide NPs, based on the relationship between the cellular redox potential to

band gap energy levels of metal and metal oxides. The authors suggest that these NPs with a diameter larger than 20–30 nm, whose band gap energy (E

_{c}) falls within the range of cellular redox potentials (−4.12 to −4.84 eV), are able to cause oxidative stress in biological systems. In a comprehensive study including

in vitro,

in vivo and multiparametric high throughput screening, Zhang

et al. [

34] confirmed this theoretical model for several metal and metal oxide NP materials. Therefore we propose a weighting factor for band gap energy levels (

w_{PC,5}). There are only very limited data which suggest that only in the selected band gap interval ROS formation occurs to a significant extent. From data cited in the literature we extrapolated a factor of 10 of the increase of weighting factor 5 as a plausible example. For NPs showing a band gap energy in the range of 4.1–4.8 eV we set

w_{PC,5} = 10, otherwise it is set to

w_{PC,5} = 1 (

Figure 5):

**Figure 5.**
Weighting factor 5 (w_{PC,5}) for the band gap energy levels of metal and metal oxides for NP larger than 20–30 nm: If NP is showing a band gap energy in the range of 4.1–4.8 eV, the w_{PC,5} is set to 10 otherwise it is 1.

**Figure 5.**
Weighting factor 5 (w_{PC,5}) for the band gap energy levels of metal and metal oxides for NP larger than 20–30 nm: If NP is showing a band gap energy in the range of 4.1–4.8 eV, the w_{PC,5} is set to 10 otherwise it is 1.

6. Proposal for the sixth weighting factor

w_{PC,6}:

w_{PC,6} is proposed to be a function of the

dissolution/dissociation rate of NPd which has been shown to be proportional to the specific surface area SSA of most NP [

35,

36,

37]. In fact, SSA is inversely proportional to the NP diameter d (Equation (1)) and proportional to the dissolution rate constant

k; the latter is NP-material dependent and also depending on the physico-chemical properties of the cytosolic or body-fluidic solvents. According to earlier reports [

38] the fractional loss of NP mass

m_{NP} over time

t is:

and the weighting factor

w_{PC,6} is proportional to the inverse of the diameter, the dissolution rate constant of the NP in the biological fluid and time

t:

Changes of the fractional rate over time are shown in

Figure 6 for NPs of different sizes and different dissolution rate constants.

**Figure 6.**
Weighting factor 6 (w_{PC,6}) is a function of the dissolution rate of the NP. Panel **a**: fractional mass loss of differently soluble NPs over NP diameters ranging from 5 to 250 nm; Panels **b**–**d** give the fractional NP mass losses over time for different NP sizes of virtually insoluble (b), moderately soluble (c) and highly soluble NPs (d).

**Figure 6.**
Weighting factor 6 (w_{PC,6}) is a function of the dissolution rate of the NP. Panel **a**: fractional mass loss of differently soluble NPs over NP diameters ranging from 5 to 250 nm; Panels **b**–**d** give the fractional NP mass losses over time for different NP sizes of virtually insoluble (b), moderately soluble (c) and highly soluble NPs (d).

#### 3.2.3. Consequences for Equivalent Dose of NP

To summarize the above mentioned model, the equivalent dose of NP (H_{NP,T}) is calculated by multiplying the deposited dose to the organ or tissue (D_{NP,T}) with the nanomaterial specific weighting factors w_{PC,i}. These factors consider the physico-chemical properties of specific NPs. The product of weighting factors w_{PC,i} is proportional to the observed biological effect, of organ or tissue T for the same deposited NP dose. For weighting factors w_{PC,1}, w_{PC,3}, w_{PC,5}, and w_{PC,6} we used the equations provided in section “Equivalent dose of NP and weighting factors” and the corresponding figures. Due to the lack of in vivo and in vitro data regarding w_{PC,2} and w_{PC,4} we have suggested sets of data and provided their plausibility in section “Equivalent dose of NP and weighting factors”. These data were used to demonstrate how the deposited dose can be influenced by the physico-chemical properties of NP leading to the equivalent dose (H_{NP,T}).

Figure 7 shows the comparison of equivalent doses of different metal and metal oxide NPs being proportional to the product of the various

w_{PC,i}. For example 5 nm Ag-NP has a substantially higher

H_{NP,T} than TiO

_{2} because the

w_{PC,6} of Ag-NP is 1,000-fold higher according to its very high dissolution rate constant and its very small diameter. CuO and Ag-NP are both rather soluble and the released ions were shown to induce oxidative stress or are bactericidal, respectively. Therefore, the product of the

w_{PC,i}—which is proportional to the equivalent dose—of both CuO and Ag-NP is determined by their high dissolution rate constants. It is interesting to see the differences of the

H_{NP,T} of hematite and magnetite form of Fe

_{2}O

_{3}. Again, the product of the

w_{PC,i} is determined by their different dissolution rate constants and sizes. As shown in

Figure 7a, the equivalent dose for Au-NP is lowest since the values of

w_{PC,2–6} are minimal, while other NP—depending on their physico-chemical properties—show higher values for one of

w_{PC,i}.

**Figure 7.**
Comparison of the “equivalent dose” described by the product of all w_{PC,i} for different NPs. The tables specify the w_{PC}_{,}_{i} for each NP of 5, 50 and 100 nm diameter: (**a**) different metal and metal oxide NPs, and (**b**) for SWCNTs and MWCNTs. Note that in (**b**) Fe-ions of the dissolving Fe-oxide NPs play a key role in the Fenton reaction seen as a main source of ROS induction in biological systems, leading to oxidative stress. The tables indicate the calculation of the “equivalent dose”. For w_{PC,3} we either presumed a zeta potential below 6 mV leading to a w_{PC,3} value of 1 or a zeta potential of >15 mV leading to a w_{PC,3} value set to 10 in the absence of detailed information.

**Figure 7.**
Comparison of the “equivalent dose” described by the product of all w_{PC,i} for different NPs. The tables specify the w_{PC}_{,}_{i} for each NP of 5, 50 and 100 nm diameter: (**a**) different metal and metal oxide NPs, and (**b**) for SWCNTs and MWCNTs. Note that in (**b**) Fe-ions of the dissolving Fe-oxide NPs play a key role in the Fenton reaction seen as a main source of ROS induction in biological systems, leading to oxidative stress. The tables indicate the calculation of the “equivalent dose”. For w_{PC,3} we either presumed a zeta potential below 6 mV leading to a w_{PC,3} value of 1 or a zeta potential of >15 mV leading to a w_{PC,3} value set to 10 in the absence of detailed information.

In case of SWCNTs, the ratio of the surface atoms to the total = 1 since all atoms are on the surface. The

w_{PC,2} namely the surface texture is set to 10 since it is known that the form of the CNT and the length are relevant for the induction of biological effects. Moreover, it is known that the length-to-width ratio (aspect ratio) of the CNT is related to the “frustrated phagocytosis”. Therefore

w_{PC,4} is the main weighting factor which is set to 500 for ≥15 µm SWCNTs (

Figure 7b), since we consider the development of mesothelioma including subsequent death (as has been proven by many asbestos workers) as the highest risk. For MWCNTs the

w_{PC,1} is different, depending of the number of layers and the correspondingly smaller ratio of surface atoms to total atoms, therefore the magnitude of the equivalent dose is 50-fold lower in this case. In summary, the application of

w_{PC,i} allows the comparison of the product of all weighting factors which is proportional to the equivalent dose of specific NP, and allows also a quantitative ranking or categorization of specific NPs.

Figure 7 provides a first attempt how the equivalent NP dose can be ranked and it shows the huge differences between the various NP due to different physico-chemical properties.