Mathematical Model

For full details of the mathematical model, please see (Martin et al., 2011). Briefly, the major buffer in the blood and tissues is bicarbonate, which acts via the following chemical reaction that is accelerated by the presence of carbonic anhydrase: Therefore, our model tracks the levels of bicarbonate, carbon dioxide, and free protons in each compartment. We adjust the reaction rate constants to incorporate the acceleration of the reaction by carbonic anhydrases present in both the blood and tumour. Additional buffering occurs in both blood and tumour compartments due to intracellular buffering, and fixed and mobile protein buffers (Hainsworth, 1986, Davenport, 1974) though these contributions act on a faster timescale than that of the HCO − 3 /CO 2 buffer. As there is little to no movement of intrinsic buffers between compartments, we assume the buffering contribution in the tumour compartment is constant and implicitly incorporate it in the tumour proton production parameter. The model tracks arterial blood delivery to the tumour, and this blood compartment contains hemoglobin in the oxygen-bound form with low proton carrying capacity. It is reasonable to assume only a small proportion of blood delivered to the tumour (that which is delivered to hypoxic areas) will contain the deoxygenated form of hemoglobin which can bind protons, and hence we neglect this small contribution as a first approximation. The hypoxic tumour subcompartment would be low in bicarbonate, high in CO 2 , and likely to have poor flow and connectivity to the vascular network, with the latter which most likely reduces the potential efficacy of any buffer delivery to that region. Subsequently, our model could be extended to incorporate additional buffering components at different spatial and temporal scales. Although it is a first approximation, the model has been validated against known data to ensure accuracy (Robey et al., 2009).

The model is formulated in the blood and tumour compartments, with the chemical reactions, vascular exchanges, physiological responses, and treatment terms. Here, B t,b represents the bicarbonate in the tumour and blood respectively, H t,b the free protons, and C t,b the carbon dioxide.
1.1 The ratio of advection to diffusion in transport across capillaries To ascertain which transport mechanisms the model should consider, we need to assess the relative importance of diffusion compared to advection for transport across capillaries, which is measured by the Peclet Number, P e . In the context of microvascular permeability this is given by (Jain, 1987): Here ∆p and ∆Π are the hydrodynamic and osmotic pressure drops across the capillary, with σ and σ F representing the osmotic and solvent drag reflection coefficients, respectively, which are constrained between zero and unity. Note that we have used the modulus of ∆p − σ∆Π in the above to ensure that P e > 0 , as is typical in the engineering and physical literature.
When P e is small compared to unity, transport is dominated by diffusion: hence to demonstrate that biophysical measurements indicate the movement of bicarbonate across capillaries is diffusively dominated it is sufficient to consider the upper bound of P e on the right hand side of inequality (2).
Estimating the above contributions to the Peclet number is difficult due to tissue variation so we consider typical scales for a single tissue to give an indication of its typical size.
The parameter grouping |∆p − σ∆Π| varies between 5-10 mm Hg from the arteriolar and the capillary to its venous end for a reflection coefficient σ = 1 (Brandis, 2011). Reducing the reflection coefficient acts to reduce this variation though the most extreme tumour interstitial pressure measured is 23mm Hg (Jain, 1987) (Jain, 1987); the hydraulic conductivity is also generally anticipated to increase, though no detailed measurements are available (ibid ). In the absence of further information, we assume any increase in hydraulic conductivity within tumours does not swamp the increase in vessel permeability, which is consistent with the conclusions that small hydrophilic molecule transport is diffusion dominated for the special case of brain tumours (Jain, 1987, Groothuis et al., 1982. As such, to the extent that the data allow conclusions to be drawn, bicarbonate cross capillary transport appears to be diffusion dominated. In the absence of detailed quantitative information on hydrogen ion permeabilities and noting that small positive ions (Na + , K + ) have a slightly enhanced diffusive permeability relative to bicarbonate in frog muscle (Jain, 1987), we hypothesise by inheritance that hydrogen ion transport across capillaries is also diffusion dominated, even in tumours.
It is reasonable to assert that carbon dioxide transport across capillaries is diffusively dominated: otherwise extensive regions of the capillary bed could not remove excess carbon dioxide during normal metabolism. Capillary diffusive permeability measurements for carbon dioxide are lacking, though these are anticipated to be roughly the same as those measured for lipid bilayers (Geers and Gros, 2000), which are four orders of magnitude larger than capillary permeabilities of bicarbonate (Jain, 1987, Gutknecht et al., 1971, quantitatively demonstrating that carbon dioxide transport is diffusively dominated.
Consequently all model constituents are taken to be subject to diffusively dominated transport across capillaries and the fluxes coupling the tumour and blood compartments of the model are driven by concentration differences, independently of interstitial pressures, by inheritance from diffusive transport.

Building the Model: Tumour Compartment.
The acid-base dynamics in the tumour compartment are: with all concentrations in mol/L, and initial conditions set as the normal arterial blood values, C t (0) = c 0 , B t (0) = b 0 , and H t (0) = h 0 . Here, γ 1 , γ 2 , γ 3 are the vessel fluxes for bicarbonate, lactate, and carbon dioxide respectively. These are given by where V AD is the vessel length per tumour cross section area (in cm/cm 2 ), and P i is the vessel permeability (in cm/s) for the respective ion or molecule (Jain, 1987). The φ terms are the tumour production terms of protons and carbon dioxide.
We assume that the hydration and dehydration reaction rates (k 1 and k 2 ) for the HCO − 3 /CO 2 conversion are equal in the blood and the tumour. In the blood, carbonic anhydrase (CA) II in red blood cells can accelerate the hydration reaction 50,000 to 1,000,000 fold over the uncatalyzed rate at human body termperature Chegwidden and Edwards (2000). Tumour associated carbonic anhydrases include CA II and CA IX (Nordfors et al., 2010, Chia et al., 2001; the activity of CA IX has recently been found to be as high as CA II (Hilvo et al., 2008). Hence, we assume for simplicity the catalytic rates in the blood and tumour are equal, though a previous analysis indicates the model is robust to changes of several orders of magnitude in k 1 and k 2 , provided the ratio of the reaction rates, and hence pKa, remains equal (Martin et al., 2011).

Building the Model: Blood Compartment.
The acid-base dynamics in the blood compartment are: The initial conditions are set as the normal arterial blood values, In these equations, v T represents the volume fraction of the tumour/blood, which varies as the tumour grows, but will be considered constant in these simulations as that is an appropriate assumption over our time span.
The model incorporates renal filtration of blood bicarbonate via the φ 2 and λ 1 terms, a detailed explanation can be found in (Martin et al., 2011). The amount of bicarbonate lost from the bloodstream to the kidney is proportional to the blood concentration of bicarbonate and φ 2 , the glomerular filtration rate (GFR). The GFR is a combined rate of the amount of bicarbonate filtered from the blood in all of the nephrons in the kidney. The rate of bicarbonate absorption is equivalent to the rate of net total acid excretion λ 1 , via the splitting of blood CO 2 by intracellular carbonic anhydrase. Although bicarbonate re-absorption in the nephron is a complicated process involving several other ions, this type of mathematical representation is commonly used in calculating acid/base disturbances (Hainsworth, 1986, Davenport, 1974. The terms φ 3 and φ 4 represent the contribution from the rest of the body tissues into the blood of protons and carbon dioxide, respectively.
1.4 Building the Model: Ventilation.
In the system, we construct a ventilation term where CO 2 lost through ventilation is proportional to the product of the ventilation rate and the CO 2 concentration. The function for ventilation rate is approximately linear with minimum and maximum thresholds (Widdicombe and Davies, 1983). This curve has been well quantified experimentally in both humans and mice (Mitchell and Singer, 1965, Fencl et al., 1969, Yee and Scarpelli, 1986. Essentially, CO 2 lost through ventilation is proportional to the product of the ventilation rate, f (C b ), and the CO 2 concentration. The function for ventilation we use is: We neglect the effect of H + on ventilation rate as the presence of respiratory compensation to metabolic alkalosis (our examined state) is controversial, often not present in humans and dogs, and when present the magnitude of compensation is highly variable and in all cases limited to a low level , Javaheri et al., 1982, Hornick, 2003, Feldman and Zimmerman, 2001).

Sensitivity analysis
The calculation of a sensitivity coefficient allows the quantification of the effect a change in a parameter, p, has on one of the variables, V . This can be calculated by the equation, allowing the identification of parameters which have the most effect in altering the tumour pHe, as well as how the treatment term can affect the pHe of the tumour and blood. The full derivation and results are presented in (Martin et al., 2011), but a subset of these results and clinical implications are discussed in this manuscript.

Hypothetical exogenous buffer
An important treatment alternative is the option to use an alternate buffer to sodium bicarbonate, or a combination of bicarbonate and another buffer. Therefore, we extend the previous model to include an additional non-volatile hypothetical buffer. With this extended model, it is possible to explore the ideal characteristics of a hypothetical buffer, thereby highlighting the buffers with the most treatment potential.
Here we extend our system by modelling the addition of a hypothetical buffer solution which contains the buffer in its conjugate base form, A − , and its acid, D, which react in the following manner: Both the acid and its conjugate base can diffuse from the blood into the tumour. In the body there are organic and inorganic proteins which act in this way, although they are at such low concentrations that they have only a minimal effect on blood buffering. However, we would like to know if a treatment involving an alternative buffer with a different pK a would have a better clinical outcome, and if so, what pK a or buffer to try in future experiments.
As before, B t,b represents the bicarbonate buffer in the tumour and blood respectively, The tumour equations are as follows: The blood equations are as follows: with initial conditions, As in the previous model, γ 4 and γ 5 are the vessel flux rates. The treatment terms are θ 6 and θ 7 , and λ 3 and λ 4 represent general loss terms, such as due to kidney filtration. The actual physiology of this term may differ depending on the particular buffer, but this general loss term would still be appropriate. Importantly, we explore two different types of treatment in this section: untitrated and titrated. If the treatment is untitrated, then the total dose of buffer (M/L/sec), TotDose, is administered in the A form. Hence, θ 6 =TotDose and θ 7 = 0. If the treatment is titrated then H + is added to the A form to create a solution of A and D at any desired pH. Therefore, if the treatment is titrated to the blood pH of 7.4, then θ 6 = 1 − 1 1+10 7.4−pKa ×TotDose, and θ 7 = 1 1+10 7.4−pKa × TotDose. The advantage of this is that a buffer can be taken with any pK a , and if it is titrated to the blood pH of 7.4 it will not change the blood pH when administered as a treatment. Instead, it will just increase the concentration of buffer in the blood, increasing the buffering capacity.
The ventilation term, f (C b ), remains the same as in our original model, and the human parameter values used are in Table 1.
We assume that the hypothetical buffer is similar in size to HCO − 3 , and therefore has approximately the same vessel permeability (γ 4 = γ 5 = γ 1 ). In reality, size, charge, and solubility will all affect its delivery to the tumour. As we assume the buffer is completely exogenous, the initial conditions of each are zero. The ratio of k 4 and k 3 is determined by the pK a we choose. As the specific values of k 3 will differ among buffers, we estimate that the reactions proceed on the same timescale as the bicarbonate reactions, and let k 1 = k 3 and from the equation for pK a , calculate k 4 = k 3 10 −pKa . The values of θ 6 and θ 7 are determined by our initial dose and titration. If the buffer is solely administered in the A form, then θ 7 = 0. If the solution is titrated to the blood pH, then the total dose is split between the D and A forms. We assume that the buffer is lost through renal filtration in the glomerulus, therefore λ 3 = λ 4 = λ 1 as GFR is constant regardless oft he exogenous buffer molecular size.

Name
Value Units k 4 k 3 10 −pKa 1/s k 3 k 1 L/mol ×s γ 4 γ 1 1/s γ 5 γ 1 1/s λ 3 λ 1 1/s λ 4 λ 1 1/s  (Martin et al., 2011) for full details and parameters). Red squares represent inducing a metabolic disturbance by varying HCO − 3 with a constant pCO 2 level (40 mm Hg). Red triangles represent the effect of varying CO 2 through disordered ventilation rates. These data points were obtained by fixing the ventilation rate at several values, running the simulations and taking the blood CO 2 , HCO − 3 , and pH values prior to renal compensation consistent with experiments. Following the pCO2 isopleth, the model produces an excellent fit, and falls within normal limits for the in vitro blood buffer line. In comparing the model predictions to the in vivo data, the model falls within the 95 % confidence limits of the experimental data, particularly within the biological pH range of primary importance (7. 35-7.45). Only at very acidic pH is there a deviation from the predicted buffer line, which is acceptable as the model is primarily focused on the potential creation of metabolic alkalosis, not acidosis. Deviations from in vitro and in vivo measurements are likely due to electrolyte distribution and different buffering capacities of cells and interstitial fluid. Reproduced from (Martin et al., 2011).

Sensitivity coefficient
Blood H+ Tumour H+ Figure 2: Human sensitivity coefficients with an absolute value greater than unity, with treatment, θ 1 = 6 × 10 −7 . The magnitude indicates how sensitive the tumour and blood proton concentrations are to a particular parameter, with larger magnitudes indicating more sensitivity. Notably, the tumour proton level is most sensitive to the parameters involved with renal function Φ 2 and ξ 1 , but the blood proton level is also highly sensitive to changes in these as well. Alternatively, the tumour proton concentration is also sensitive to parameters involved with tumour proton production (Φ 1 ) and ventilation (∆ 1 and ∆ 2 ), and the blood proton level is not as sensitive to changes in these.