# Metrics, Dose, and Dose Concept: The Need for a Proper Dose Concept in the Risk Assessment of Nanoparticles

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## Abstract

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## 1. Introduction

- (1)
- The deposited dose, which is the total deposited NP surface area (SA) per tissue mass or volume (m
^{2}/kg, or m^{2}/m^{3}or m^{−1}). This SA of the NP has the potential to induce biological effects. Moreover, we consider agglomerated NPs as well. - (2)
- The equivalent dose, whereby the deposited NP dose is weighted by factors quantifying the effects of several other physico-chemical properties of the NPs, such as the specific surface area, surface texture, electron band gap interval at the NP surface, surface charge (zeta-potential), NP morphology (shape, surface roughness, length-to-width ratio (aspect ratio)), and the dissolution rate.

## 2. Dose Assessment for Ionizing Radiation

#### 2.1. Exposure to Ionizing Radiation

#### 2.2. Absorbed Dose of Ionizing Radiation

^{−1}or the special unit gray (Gy).

_{T}, which is the mean dose to a target organ or tissue T, is often used. For internal exposure D

_{T}is expressed by the committed dose to T for a defined time period τ (taken to be 50 years for adults). Then:

_{T}(t) is the mean absorbed dose rate in T at time t (Table 1).

Dose quantities | Radiation | NP |
---|---|---|

Absorbed/deposited dose (D): | The absorbed dose (D) is the mean energy dε imparted to matter of mass dm by ionising radiation, J·kg^{−1} or Gy: | The deposited dose of NP (D_{NP}) is the total deposited surface area (SA) of NP per mass of living matter dm, m^{2}/kg: |

Dose rate: | The mean absorbed radiation dose rate in T at time t: Ḋ_{T}(t) | The mean absorbed NP dose rate in T at time t: Ḋ_{NP,T}(t) |

Committed tissue dose: | The quantity of radiation absorbed per unit time: | The quantity of absorbed NP dose (uptake) per unit time: |

Equivalent dose: | The mean absorbed dose from radiation R in a tissue or organ T, multiplied by the radiation weighting factor w_{R}, J·kg^{−1} or Sv: | Absorbed dose weighted by NP property(material) dependent reactivity weighting factor(s) w_{PC}_{,i} (i = 1, 2, ….) for different physico-chemical properties or functional “behaviors” of NP: |

Effective dose: | The tissue-weighted sum of the equivalent doses in all specified tissues and organs of the body, J·kg^{−1} or Sv: | The tissue-weighted sum of the equivalent doses in all specified tissues and organs of the body: |

#### 2.3. Equivalent Dose of Ionizing Radiation

_{T}), whereby the quality of radiation is adjusted by dimensionless radiation weighting factors [24,25]. These factors were determined by comparing the induced biological damage of the radiation to that with an equal dose of gamma or X-rays. X-rays and gamma rays have a weighting factor of 1 (1 Gy = 1 Sv), whereas e.g., for the same absorbed dose, alpha particles are considered to be 20 times biologically more potent for stochastic effects [24,25]:

_{R,T}is the absorbed dose from radiation R in a tissue or organ T, and w

_{R}the radiation weighting factor.

#### 2.4. Effective Dose of Ionizing Radiation

_{T}.

## 3. Dose Assessment for NPs

#### 3.1. Exposure to NPs

#### 3.2. Dosimetry for NPs

#### 3.2.1. Deposited Dose of NP

_{NP}) is the total deposited surface area (SA) of NP per mass of living matter (m):

_{NP}(t) is one more important variable (or entity) which has to be taken into account [29]. Factors affecting the dose rate include the biodegradability of certain NPs. For highly biodegradable NP the time for biological response is relatively short, whereas other NPs may not be degraded within the body. In addition, fractions of NP, depending on their physico-chemical properties, may be retained in the organ where they originally were deposited after intake or they may be excreted, whereas others may translocate into the circulation for biodistribution and subsequent accumulation in secondary organs or for excretion. In analogy to the absorbed dose in radiation dosimetry, the deposited NP dose in a given organ or tissue, D

_{NP}

_{,T}is adopted. This is the cumulated mean dose to a target organ or tissue T. The dose D

_{NP}

_{,T}is expressed by the committed dose to T for a defined time period τ. Then:

_{NP,T}(t) is the mean absorbed dose rate in T at time t (Table 1).

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

_{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.

_{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:

_{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.

_{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.

_{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.

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

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

_{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:

_{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.

_{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:

_{PC}

_{,1}becomes a constant:

_{PC}

_{,1}= f (SSA

_{pp}) = c

_{agg}for agglomerates of same-sized primary particles

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

_{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.

_{pp}w

_{PC}

_{,1}is also a constant c’

_{agg}depending solely on the mean SSA

_{pp}of the primary particles (Figure 1c,d)

_{PC,1}= f

**(**SSA

_{pp}

**) = c’**for agglomerates

_{agg}_{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.

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

_{PC,2}:

_{PC,2}~(σ‘/σ

_{0}‘)/d with 1 < σ‘/σ

_{0}‘ < 10

_{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.

_{PC,3}(arbitrary value):

_{PC,3}= max{1, 1.12 × ZP − 6.23}

**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.

_{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):

_{PC,4}= 1 for NNI at normal phagocytosis

w

_{PC,4}= 500 for NNI at frustrated phagocytosis

**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.

_{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):

_{PC,5}= 10 for NP atoms/molecules within band gap interval of 4.1–4.8 eV

w

_{PC,5}= 1 for NP atoms/molecules outside the above band gap interval

**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.

_{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:

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

**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

_{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}).

_{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.

_{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.

#### 3.2.4. Effective Dose

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Simkó, M.; Nosske, D.; Kreyling, W.G. Metrics, Dose, and Dose Concept: The Need for a Proper Dose Concept in the Risk Assessment of Nanoparticles. *Int. J. Environ. Res. Public Health* **2014**, *11*, 4026-4048.
https://doi.org/10.3390/ijerph110404026

**AMA Style**

Simkó M, Nosske D, Kreyling WG. Metrics, Dose, and Dose Concept: The Need for a Proper Dose Concept in the Risk Assessment of Nanoparticles. *International Journal of Environmental Research and Public Health*. 2014; 11(4):4026-4048.
https://doi.org/10.3390/ijerph110404026

**Chicago/Turabian Style**

Simkó, Myrtill, Dietmar Nosske, and Wolfgang G. Kreyling. 2014. "Metrics, Dose, and Dose Concept: The Need for a Proper Dose Concept in the Risk Assessment of Nanoparticles" *International Journal of Environmental Research and Public Health* 11, no. 4: 4026-4048.
https://doi.org/10.3390/ijerph110404026